Shifting and widening my scope a little, here’s a look into the history of two consonant clusters across the Samic languages as a whole.
The two-glide cluster *-jv- is a simple place to start. The development of this is straightforward: this is retained essentially intact everywhere across all Sami varieties. (If you want to have a look for yourself, I am including copious links to the Álgu database in this post.) Possibly the coda *j may have been vocalized into the 2nd component of a diphthong/triphthong, but this is basically trivial.
A small further complication still comes up in Southern Sami. Two words have here a seemingly irregular *-jj-: *oajvē > åejjie ‘head, end’; and *peajvē > biejjie ‘sun; day’.
Both of these happen to be inherited Uralic words, with cognates stretching all the way to Samoyedic. So my first reflex was to go “a ha! does this mean that the words showing /jv/ are therefore newer loanwords?” The answer is “no”, though: at least *koajvō- > gåajvodh ‘to dig’ is of equally ancient pedigree. But I think I can dial this hypothesis back a little. Perhaps the shift *jv > *jj occurred due to the following front vowel *-ē (in Southern Sami characteristically diphthongized to /ie/ even in the 2nd syllable). This seems phonetically plausible & drops the number of counterexamples from half a dozen to one: *vājvē > vaejvie ‘pain’. This last word is in turn a known Finnish loanword, which may have indeed diffused into Southern Sami at a late date.
This idea seems to be preliminarily further supported by an interesting derivative of ‘head’: åajvadidh ‘to advice’. My chops in SS historical morphology are insufficient to present an implicit PS reconstruction, but we can clearly see here at least a retained stem vowel *ā, a regular feature before 3rd syllable *ë; in other positions this was further raised to *ē already in PS. And before this lower vowel, *-jv- survives after all.
Now let’s consider the opposite PS cluster *-vj-. This turns out to have had a much more complicated history.
Three Sami varieties have completely regular development. Lule Sami and Ter Sami have in all involved words metathesized this cluster, merging it with *-jv-. Inari Sami has always retained /vj/. Northern Sami also might belong here, depending on who you ask: the Álgu database claims /jv/ in a single word *sāvjë > sájva ‘isolated lake’, while my copy of Yhteissaamelainen sanasto presents sawˈjâ (= equivalent to sávja in the current NS orthography). I would guess that there are dialectal differences involved? FWIW, Sammallahti in The Saami Languages claims that at least the Torne Sami dialect group “originally” belonged with Lule Sami rather than Northern Sami. 
In a couple of other varieties, it is also possible to state a mostly applicable rule. Pite Sami aligns with its sibling Lule in having -jv- everywhere except in *jēvjë > jievja ‘white reindeer’; while Skolt and Kildin Sami align with Inari Sami in having -vj- everywhere except in *ćōvjë > Sk čuõivâk, K čuəivex ‘grey reindeer’. Probably these sorts of exceptions again represent loaning from neighboring dialects. 
Southern Sami again shows a few more complications; as does the neighboring Ume Sami. Covering SS first, metathesized /jv/ occurs in two words: *tāvjā > daajvaj ‘often’, *sāvjë > saajve ‘gnome’. Unmetathesized /vj/ is found in three: *jēvjë > joevje ‘light grey reindeer’, *jēvjë- > joevjeme ‘beard moss’ (don’t ask me what’s the oe doing in these), *vōvjē > vuevjie ‘wedge’. Lastly, an assimilated /jj/ is found in *ćoavjē > tjåejjie ‘stomach’. This appears to confirm the assimilation rule I proposed in the 1st section: v > j / j_ie. Provided that we assume the metathesis *vj > *jv to have occurred before this…
The Ume Sami reflexes seem to support this last assumption. Although not many of the involved words have been recorded from here, /jv/ is found in those lexemes that have SS /jv/ ~ /jj/: dàivài ‘often’ and tjåìvee ‘stomach’ — while /vj/ is found in those that have SS /vj/: jauja ‘grey reindeer’, vyöyjee ‘wedge-shaped patch’. There is also one word with a somewhat baffling three-glide reflex: guyvjas ‘grey reindeer’ (with unetymological /g-/ to boot). 
How should this distinction between S+U-metathesizing and S+U-unmetathesizing *-vj- be accounted for? Could this be etymological somehow? An interesting fact is that *vōvjē ‘wedge’ is one of the Samic words showing lenition of original coda *k before sonorants (as shown by the Finnic cognates: e.g. modern Finnish vaaja, Karelian voakie, Livonian vaigā < PF *vakja).  So, perhaps this change occurred only after the metathesis of inherited *vj to *jv in Southern and Ume Sami? A late date for the change has already been suspected:
This sound change cannot be reliably dated, but it may well have taken place during a relatively late phase of Proto-Saami.
(Aikio 2006: 3.11 §) 
With this interpretation, a “maximally hereditary” chronology would be:
- Lenition *kj > *ɣj in Finnish.
- Samic *tāvjā ‘frequent’ is loaned from Finnish *taɣja.
- Metathesis *vj > *jv in South & Ume.
- Lenition *kj > *vj all across Samic.
- Samic *jēvjë ‘white reindeer’ is loaned from Germanic.
- Assimilation *v > j / j_ie in South. — Metathesis *vj > jv in Pite, Lule & Ter. — Raising *eu > *iu in Germanic.
- Samic *vājvē ‘pain’ is loaned from Finnish vaiva.
…But is it a good idea to attempt maximizing the degree to which various Samic words would have been inherited from a common ancestor? I think it is important to keep in mind that fresh loanwords readily diffuse across dialect continua.
As for the particular downsides of the abov scenario, at minimum I am uncomfortable assuming that the specifically Finnish change *kj > *ɣj occurred earlier than the supposedly Proto-Germanic change *eu > *iu / _j.  OK, it’d be possible to go on making some cleanup assumptions; e.g. that in the numerous newer Germanic loans in Finnish where *ɣj can be reconstructed, this was substituted for original *kj; or perhaps, that the /k/ ~ /g/ found in the other Finnic languages would be a reversal from *ɣ; but this would all be for no other reason than ensuring a Proto-Samic ancestry for SS daajvaj, US dàivài. We could instead assume that S+U acquired these words from the direction of P+L, and show /jv/ for this reason.
This should also call into question whether my step 3 above existed at all. *sāvjë ‘gnome’ (elsewhere in Samic also with meanings like ‘underground water’, ‘lake with an underworld entrance’, ‘isolated lake’) seems like a potential cultural loan from the P+L direction at least. It is of Germanic ultimate origin, but seems to have acquired its mythical flavor only on the Sami side: the PGmc root is simply *saiwiz ‘lake’.
Note moreover that this loan etymology actually predicts PS *-jv-, not *-vj-! And yet there is no evidence for the inverse metathesis *-jv- > *-vj- to have regularly occurred in any Samic variety. So are we therefore forced to furthermore conclude that this word was originally adopted specifically in the Pite/Lule area, and hypercorrectly metathesized to *-vj- when loaned eastward from these varieties? The Southern /jv/ could similarly also turn out to be original after all.
This leaves just the question of *ćoavjē ‘stomach’. Relationship to Samoyedic *t¹äjwə ‘stomach’ has been proposed. The initial consonant, vowel frontness, and glide cluster order all fail to match, though, so I suspect this is only an accidental resemblance. I could just as well propose that the Samic word is a metathesis from something like earlier *voaćjē, and therefore related to Finnic *vacca ‘stomach’? (Ha ha.) With the case for inheritance being in this shape, I don’t think it would be too much of a problem to assume that here, too, the S+U forms have been loaned from the direction of P+L. — But still early enough to have participated in cluster smoothing in SS, apparently.
An additional topic to ponder at this point would be the motivation of the metathesis *-vj- > *-jv-, which altogether appears to be attested in at least two widely separated parts of the Samic dialect continuum. Pite and Lule Sami are spoken in northern Sweden and adjacent areas of Norway (also Finland if we count Torne Sami), Ter Sami at the eastern end of the Kola peninsula. It seems unlikely that these groups have been in any direct contact with each other since Proto-Samic times. It also seems unlikely that this incredibly specific metathesis was purely coincidentally innovated in both. One possibility might be some kind of a phonological precondition for this change having existed already in Proto-Samic, which in only two areas led to the change running to completion?
A better solution though might be a common external source. This exact same metathesis [edit 2016-07-05: a fuller blog post on this topic now available] happens to be known furthermore from the Finnic languages! Late Proto-Finnic allows no *-vj- (or *-Vuj-/*-Vüj-: we are better off reconstructing diphthongs rather than coda glides at this date), and although no words with PU *-wj- have been retained in Finnic, a number of loanwords allow reconstructing a metathesis here. E.g. PGmc *flauja- → Finnish laiva ‘ship’.  Metatheses of some other similar clusters including older *-wr- (PS *jāvrē ~ LPF *järvi ‘lake’) are also found, which suggests that this type of change originated in Finnic, and might have been in the case of *-vj- > *-jv- passed on to Samic.
Still, why just these specific varieties? The Lule Sami probably had numerous connections with Finnic traders and settlers in the Torne Valley and adjacent areas since a much older period than the Finnmark/Inari/Skolt/Kildin Sami living further inland, that much is clear. Yet should we expect this shift to have therefore also been also present in the extinct “southeastern” Sami varieties such as the marginally attested Kemi Sami?
Particularly difficult to understand is Ter Sami. I do not think we even know at present whether the Kola Sami languages developed entirely in situ, or if they may have spread to Kola from e.g. the southern reaches of the White Sea, some of their characteristic features already in tow? The presence of this sound change might demand, at minimum, for Ter to descend from a dialect that was originally spoken further south than the corresponding ancestral dialect of Kildin…
 One wonders how and why may we claim that it no longer does; or whether we are to conclude that “Northern Sami” is an areal entity rather than a genealogical one.
 I wonder if these last two words have some relation to each other. The semantic closeness is obvious, and the consonant skeletons are quite similar as well. The proposed etymology for *jēvjë is loaning from (pre-?)Proto-Germanic *heuja- ‘hue’, and the Germanic *h- moreover comes from PIE *ḱ-. Wiktionary mentions here e.g. Lithuanian šývas ‘white’. Could any of the Satemic cognates have plausibly been loaned to yield pre-Samic *ćawjə or *ćowjə ‘grey’?)
 Or could this indicate a substitution *ḱ- >*k-, from some non-Satem variety? Perhaps not, since this would be chronologically problematic and there are other known examples of irregular *ć- > *k- in some varieties of Sami.
 Also by the word’s etymology as stemming from Baltic: cf. Lithuanian vagis, Latvian vadzis. For more details cf. Itkonen, Terho (1982): Laaja, lavea, lakea ja laakea. In: Virittäjä 86.
 Aikio, Ante (2006): On Germanic-Saami contacts and Saami prehistory. In: SUSA 91.
 I actually suspect this was “only” Northwest Germanic, given how Gothic shifts *e to *i always anyway. More details to come on this point later though. At any rate this would still not be a huge chronological relief.
 For further details cf. Koivulehto, Jorma (1970): Suomen laiva-sanasta. In: Virittäjä 74.
An alternate interpretation of the history of the cluster *kj in Saami could be suggested in light of Proto-Saami *ruovjē ~ Proto-Finnic *rakja ‘limb’. Most probably Ume Saami ruakssjá ‘thigh’ ( *kš, which is phonetically quite natural ([j] > [ç] > [š́] after an unvoiced consonant). Thus, Proto-Saami *kj could be reconstructed distinct from *vj; but this would imply that South Saami vuevjie, Ume Saami vyövjje ‘gusset’ are loanwords.
Again, wordpress garbled up my post, so that some text went missing. So, this Ume Saami shows the development kj > kš. The word is missing in Álgu, but I think it is mentioned by Sammallahti in “The Saami Languages: an Introduction”, in the appendix on historical phonology.
Interesting paper! I’ll take this opportunity to talk to the author about section 3.1. :-) As an Austrian, I can confirm that the expected form das Fut still exists in Sufficiently High German, although it’s very rarely used because the curses here strongly tend to be scatological (like in Czech) rather than sexual (like in English, Hungarian and Slovak). Die Fotze also exists, but has a completely different meaning over here, namely “bitchslap”. ~:-| I wonder if, in the northern meaning anyway, this word once was a n-stem nickname that underwent Kluge’s law (Google Books link to other examples).