The Proto-Indo-European laryngeals seem to form, in most people’s thinking, a kind of a phonological subsystem. Usually they end up as a class of back fricatives, or at least some kind of weaker back consonants. They certainly have similar diachronic behavior… but if this implies also unique synchronic similarity is not immediately obvious. After all, there is a rather wide range of consonants that can be easily lost from a language (in the “merges with zero” sense). And inversely: even if many members of some natural class are lost, not every one of them will have to. E.g. transient voiced spirants in various Uralic languages: early pre-Permic *β *ð *ɣ are all lost by late Proto-Permic, while out of late Common Finnic *β *ð *ɣ in Eastern Finnish/Karelian, only the latter two are lost, and *β instead gives /v/.
Occasionally PIE internal reconstructors will go further still, and point out that the most widespread reconstruction with three laryngeals would be tempting to compare with the three series of velar consonants, suggesting rewriting *h₁ *h₂ *h₃ as *x́ *x *xʷ. The analogy is clearly imperfect though. E.g. the laryngeals do not show much signs of a centum / satem isogloss, not along the usual dividing line at least;  there are no parallels to the conditional neutralizations among the velar stops, such as *ḱr > *kr; the labiovelar stops *kʷ *gʷʰ *gʷ do not show any *o-coloring effects (for *k *gʰ *g some *a-coloring effects have been proposed though). A more common objection still however seems to be that there is a widely held alternate hypothesis: many mainstream IEists think that *h₃ is better mapped as a voiced fricative: [ɣ], [ʁ] or [ʕ], and *h₁ as a glottal consonant: [h] or [ʔ].
This semi-consensus view still assigns *h₂ as a voiceless back fricative: [x] or [χ], as the direct Anatolian evidence also strongly suggests. The occasionally suggested pharyngeal [ħ] can be IMO ruled out per arguments such as those in Michael Weiss’ recent paper. (I have already opted to use *x and not *h₂ in my index of the LIV roots, and will mostly do so in the rest of this post too.) However, this leaves an opening for an objection that does not seem to be commonly made, but to me feels quite relevant. If *h₁ and *h₃ are really something like *h and *ɣ, would *h₂ = *x then really be an isolated voiceless velar fricative, without palatovelar and labiovelar counterparts? 
A brief typological survey shows that such gaps among back fricative systems are indeed not common. In particular, any language that has both /kʷ/ and /x/ is rather likely to also have /xʷ/.  A look at the PHOIBLE data turns up the following results:
- all of /k kʷ x xʷ/: 35 languages
(Bilin, Buwal, Central Atlas Tamazight, Central Siberian Yupik, Chipaya, Chipewyan, Comox, Cupeno, Dghwede, Gavar, (Paraguayan) Guarani, Gwandara “4 and 6”, (Northern) Haida, Iraqw, Jicarilla Apache, Kumiai, Lagwan, Lamang, Luiseno, Mezquital Otomi, Nootka, Quileute, Seri, Serrano, Shuswap, Tachelhit, Tera, (Southern) Tiwa, Tlingit, Tolowa, Tonkawa, Wamey, Wichi Lhamtes Nocten, Yuqui)
- only /k kʷ x/: 14 languages
(Awing, Ese Ejja, Kwasio, Nizaa, Nuclear Daba, Purepecha, Saliba, Sui, Taushiro, Tilquiapan Zapotec, Uru, Ute-Southern Paiute, Yala, Yurok)
- near misses: Haka-Chin with /k kʷ x w̥/, Izi-Ezaa-Ikwo-Mgbo with /k kʷ χ/, Wuzlam with /k kʷ χ hʷ/.
So a language that has /kʷ/ and /x/ is about 2.5 times more likely to have /xʷ/ than not; a very substantial result, when otherwise only some 3.2% of the languages in the
world PHOIBLE sample have /xʷ/.
There are moreover plenty of languages that have /k kʷ/ and some non-velar pair of ±labialized back fricatives. The most popular setup by far is /k kʷ h hʷ/ (Amharic, Arabela, Argobba, Cherepon, Fwe, Gikyode, Guinea Kpelle, Gwandara “2”, Hausa, Ikwere, Inor, Iyive, Kamayura, Kawaiisu, Kistane, Mbuko, Merey, Mesqan, Mofu-Gudur, Moloko, Nuclear Igbo, Piaroa, Sebat Bet Gurage, Siona, Suya “2”, Vame, Wandala, Wari, Wolaytta, Yeyi). Now, /k kʷ h/ is also very common; but given that x > h is a common sound change, it seems likely that many of this group of languages have come about from earlier *x *xʷ. In three cases /h hʷ/ also combines with an unpaired buccal back fricative: /k kʷ x h hʷ/ (Mfumte, Nyam), /k kʷ χ h hʷ/ (Tewa).  Other similar inventories are:
- /k kʷ ç çʷ/ (Quechan)
- /k kʷ χ χʷ/ (Bana, Kabyle, Xamtanga)
- /k kʷ ħ ħʷ/ (Bade)
- /k kʷ χ χʷ ħ ħʷ/ (Moroccan Arabic)
Lastly there is also the notable Pacific Northwest cluster of languages (Bella Coola, Coeur d’Alene, Lushootseed, Spokane, Squamish, Straits Salish, Upper Chehalis) with either just /kʷ xʷ/ (no plain velars; all have non-labialized uvulars though) or /k kʷ xʷ/ (with /k/ looking like a recent reintroduction by loans). This is tangential to the question, though.
Remarkably, this typological trend continues even within Indo-European! Nowadays Hittite is analyzed as having indeed phonemic /xʷ/ ḫu ~ uḫ beside plain /x/ ḫ (for a recent detailed review see Suter (2014) ). Per correspondences like Lycian /kʷ/ q, the same is also thought to have been the case already in Proto-Anatolian. This *xʷ corresponds to traditional PIE *h₂w, and is usually considered to come about by simple cluster coalescence. It would be however also quite feasible to set up *xʷ already for PIE itself, so that there wouldn’t be any asymmetry in stop versus fricative labialization. (This idea is supported already by Suter, whose article I only found after coming up with the idea myself.)
This will require a slight change in thinking: the concepts of “laryngeal” as “a consonant that is deleted” and “laryngeal” as “a back fricative” will need to be uncoupled. *xʷ will be a “laryngeal” in the second sense, but not in the first: it leaves at minimum a *w behind in core IE, after all. I think this sharpening of concepts would be beneficial, as Indo-European studies already suffers from treating the laryngals as excessively phonetically vague.
I belive additional evidence for *xʷ can be also found in PIE root structure. Clusters of (plain) velar + *w are often set up for PIE, but they’re much rarer than the labiovelars proper. LIV has the following counts: *kʷ 15 + 18 (root-initial + root-final), *gʷʰ 8 + 14, *gʷ 17 + 16; — *kw 7 (root-initial only), *gʰw 0, *gw 2. For *xw there are however 18 cases initially + 7 finally, which would make this both the most common *Cw cluster and by far the most common *HR cluster in PIE. 
Even more interesting are the verb root *xwyedʰ- ‘to strike dead or injured’, and the noun *xwl̥h₁néx ‘wool’: these appear to have a very rare *CRR- onset structure, unparalleled elsewhere in PIE to my knowledge. Reconstructing a monophoneme *xʷ and not a cluster **xw would however reduce these to the usual *CR-. Labiovelar stop + resonant clusters are rare as well, but at least attested, e.g. *kʷles- ‘to furrow’, *gʷyeh₃- ‘to live’, *gʷʰreh₁- ‘to smell smth’.
I would even suggest that some further internal reconstruction can be applied here. The typical onset structure in PIE is *(F)(T)(R)- (with F = fricatives, T = stops, R = resonants). In traditional reconstructions this is however violated by a number of cases of *w + resonant (attested in LIV: *wl- 1–3, *wr- 10–12, *wy- 3). However, many of these could be probably replaced by *xʷR-. Even the development to attested /wr-/, /vr-/ in a few descendants such as Germanic and Indo-Aryan would not have to be common core IE: it could represent independent developments, versus direct loss (or maybe *xʷ > *x > *h > ∅) in branches like Italic. — For *wy- some cases seem to be attested almost solely in zero-grade. They could probably be also reconstructed with *i as an original non-zero-grade root vowel, and an analogical full grade in some sporadic Indo-Iranian reflexes, similar to the case of *bʰux- ‘to grow’.
The above is just structural reanalysis, so far. It is less clear to me so far if setting up a PIE *xʷ will have implications also for the routing of the reflexes in the daughter languages; if some cases will regardless have to be retained as a cluster *xw; or even, if this could also be set up in a few additional positions.
Suter proposes one readjustment of this type: reconstructing ‘to wash’ as *lexʷ-, and not anything like *leh₃w- or *lewh₃- (and with intervocalic *-xʷ- > -[ɣʷ]- in Hittite, same as with plain *-x- > -[ɣ]-). This promisingly enough seems to cut out some ad hoc “laryngeal metathesis” rules. However, it also suggests an odd property for *xʷ: a-coloring in Latin (lavō) but o-coloring in Greek (λοέω).
How does this fit together with the seven examples I mentioned that have already been earlier reconstructed as *Ceh₂w-?
- *deh₂w- ‘to roast on a spit’: Sanskrit dunóti < *du-ne-H-, Greek δαίω, δέδηε < *daw-ye-, *de-dāw-, OHG †zuscen < *du/ū-sḱe-, Irish dóïd < *do/ōw-eye- etc.
- *geh₂w- ‘to be glad’: Greek γαίω, γάνυμαι < *gaw-ye-, *ga-n-u-, Latin gaudeō < *gāwedʰ-, and perhaps also some reflexes that LIV splits as a separate root *geh₂dʰ-.
- *ḱeh₂w- ‘to set on fire’: Greek *kaw-ye-, *kāw-s-, Lith. kūles ‘Brandpilze’ (?!), Albanian than ‘to dry’ < *ća-, Tocharian *kaum ‘sun’. Kind of a weak-looking semantic grab-bag root etymology.
- *keh₂w- ‘to hit’: reconstructed in LIV with *-h₂w- per Tocharian *kɐw- : *kåw- < *kəw- : *kāw-, even though most reflexes (Latin, Germanic, Balto-Slavic, Greek) point instead to *kuH- : *kewH-. If ad hoc metatheses are going to be assumed, why not in Tocharian rather than in all the other languages?
- *kleh₂w- ‘to cry’: Greek + Albanian *klaw-ye-.
- *melh₂w- ‘to grind’ — probably not with *xʷ, but rather an extended stem *melx-w/u-, from the more common *melx- ‘to grind’.
- *peh₂w- ‘to stop, finish’: only Greek πάυω < *paw-.
It seems that the behavior here is rather different from the ‘wash’ case, with several examples confirming a-coloring in Greek. But they also all seem to involve more complex constructions; maybe the difference could be one between coda *xʷ (retained until a-coloring?) and medial *xʷ (leniting to *w earlier?). Many also seem to involve reflexes that point to *CuH-, instead of expected *ā(w) : *aw from *ex(w) : *əx(w). And does dunóti involve o < *aw < *exʷ, maybe coming about by some kind of a *dux- > *duxʷ- development?
Nowadays lengthened grades are usually thought to be secondary, so I even wonder if instances of ā that surface here are that, instead of from *aH < *ex. The (partial) late PIE ablaut scheme for roots in *xʷ would then be *āw : *aw : *u (lengthened grade : *e-grade : zero grade). Eichner’s Law (*ēx > *ē and not **ā) on the other hand still seems to require that a-coloring is usually younger than the rise of lengthened grade.
Latin lavō can be of course also explained through Thurneysen-Havet’s Law: *o > a / _wV́. And so, if this and λοέω are *o-grades after all, there will be no trouble in assuming that *xʷ is leftwards a-coloring, just like plain *x.
So far, in summary: introducing *xʷ gets rid of several typological-phonotactic anomalies in PIE. These include at least all *CRR- roots, a large group of *CeCR- roots, possibly numerous *RR- roots, the strange abundance of the cluster *h₂w, and the unusual /k kʷ x/ inventory.
The second of these issues is, however, not exhausted by this reanalysis. CeCR- roots also regardless remain like a suspicious feature of laryngeals in particular: there are no roots in anything like **-sw-, **-dy-, only things like *-h₁w-, *-xy-. One can wonder if *xʷ is maybe only the top of an iceberg, and also a few additional “laryngeals” of this kind (back fricatives that do not get deleted entirely) should be assumed.
But there will be many other options available too, especially with laryngeals other than *x that cannot be easily grounded in direct Anatolian evidence. For very quick offhand speculation for the sake of example… since laryngeals’ presence is in some ways easier to determine than their exact position, and since in particular *-Hy- clusters are often assumed to be subject to metathesis, we could rewrite these as the more typical *-wH-, *-yH-, and simultaneously then rewrite the roots currently reconstructed as *CewH-, *CeyH- as being instead “close-vowel roots” *CuH-, *CiH- (with ablaut only secondarily by analogy).
 Thanks to various members of the Zompist Bulletin Board for a number of discussions on this topic.
 It is true that *h₂e > *a and *h₃e > *o merge often, and conceivably this could even have gone through an early merger of *h₂ and *h₃. But this happens also in the non-satemic Germanic, while failing in the satemic Armenian. The corresponding “centum” merger of *e and *a as distinct from *o also seems to be unattested entirely.
 The same could be asked of *h₃ as *ɣ too, but there happens to be a very easy answer here — just identify “missing” *ɣ́ and *ɣʷ with the semivowels *y and *w, or at least assume that the fricatives merged with the semivowels at some early stage.
 The situation for palatalized velars seems similar, but the controversy over if if *ḱ was [kʲ], or if *ḱ *k were perhaps instead [k q], makes this question harder to survey.
 How these cases have come about seems harder to figure out from just general principles. Some hypotheses I can think of would be asymmetric debuccalization, i.e. *x ≡ but *xʷ > hʷ; and later secondary lenition, such as *q > χ or *ɸ > x, some time after the introduction of contrastive labialization. Loanword phonemes could be involved, too: for a not quite exact parallel, Udmurt has /k kʷ/ natively (the latter is usually, but IMO unconvincingly, analyzed as a cluster) versus /x/ only in recent loans from Russian.
 He also refers to the same typological sound inventory argument as I do, but working with an earlier stage of PHOIBLE, he only gets together 25 examples of symmetric /k kʷ x xʷ/ versus 11 of asymmetric /k kʷ x/.
 The other *H + glide clusters come in at *h₁w at 7 + 4, *h₁y at 1 + 1–5 (with lots of cases where it seems to be unclear if *y is a part of the root that gets deleted, or a widespread suffix), *xy at 0 + 1–6, *h₃w at 2 + 2, *h₃y at 1 + 0. All *H + liquid or *H + nasal clusters occur initially only, with *xm- the most common at 7 examples. Other *Cw clusters are likewise root-initial only: *sw- 21 (in this position more common than alleged *xw, but not altogether), *tw- 8, *dʰw- 7, *dw- 5, *ḱw- 5, *ǵʰw- 3–4, *ǵw- 1–2.