We observe the following peculiar apparent relations between the conjugation patterns of 形容詞 and 動詞 (using 赤し and 行く as examples):
連体形: 赤き / 行きし (past recollective)
終止形: 赤し / 行きき (past recollective)
連用形: 赤く / 行き
連体形: 赤き / 行く
In both of these cases we have an apparent reversal of two endings across the two parts of speech: き←→し and く←→き.
Is there any explanation/etymology behind this, or is this thought to be a coincidence?
There is no clear-cut etymological explanation, but some think there is a connection. In A History of the Japanese Language (2010), Frellesvig says:
The suffixes which attach to the infinitive [i.e. renyokei] are [...] transparently agglutinating and their use as suffixes seems to be younger [than suffixes attaching to the mizenkei, which Frellesvig argues were originally suffixes beginning with /a-/ and attaching directly to the verb stem]. It is possible to view all of them as being derived from a few elements which are also reflected in a number of other grammatical morphemes in OJ. They fall into two groups: (a) forms in k ~ s, and (b) forms in t ~ n. It is likely that both sets reflect earlier copulas which were morphologized.
The k ~ s forms involve both of the past tense auxiliaries, -i(ki) and -(i)kyeri (of which the modal past seems to be derived from the simple past: kyer- < ki-ar-). The forms of the adjectival copula [by this he means endings like /-ki/, /-si/, etc.] overlap to a large extent with the past tense auxiliaries [...] This suggests that these are different morphologizations of the same (copula) material, and lends further support to the analysis of the adjectival endings as a restricted copula. Note also that the k ~ s alternation is exhibited by the adjectival copula infinitive -ku and the * -su which takes part in formation of the innovative negative forms [here he means /-zu/ and so on, which he derives from /-ni.su/] [...] The morpho-syntactic similarities between the adjectival infinitive and the negative infinitive are easier to explain if, as suggested here, they originate in variant copla forms. It is further possible that the verb se- "do", the focus particles so and ka [...] and the demonstratives ko and so [...] are root-related to these forms.
(Bold mine, everything in [square brackets] mine.)
So, there you go: a grand theory which does indeed link five of your six data points (adjectival /-ki/, /-si/, and /-ku/ + verbal simple past auxiliary /ki/ and /si/) together, albeit hand-wavily.
It's important to note that this is all speculation; by the time people started writing things down in OJ, any root-relations were long in the past. But when it comes to proto-Old Japanese, very little speculation is as informed as Frellesvig's.
Your final data point, the similarity between the adjectival infinitive ending /-ku/ and the adnominal verb ending /-u/ (as dainichi says, the ending in question only happens to be /ku/ in verbs with a stem ending in /k/, like /ik.u/) doesn't fit into this pattern at all. Since the /k/ is a red herring, really the only similarity is that it's a one-mora ending with the vowel /u/. But note that for vowel-stem and irregular verbs the ending is /-(u)ru/; meanwhile, in Eastern Old Japanese attributives sometimes end in /-(w)o/ instead of /-u/, but the adjectival /-ku/ ending never changes to an o-form. So even the vowel similarity isn't all that strong. There doesn't seem to be much of a case for a connection there.
Edit: Wait, I forgot one other non-repeated data point: infinitive /-i/ for verbs vs adnominal /-ki/ for adjectives. Here, too, though, it boils down to a single vowel -- and if you reject the connection between /-u/ and /-ku/, there's not even a semantic connection. Nothing to explain, in other words, except maybe a very general "How come so many OJ verbal/adjectival endings end in /i/ or /u/?"
