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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?
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etymology conjugations classical-japanese
Apparent reversals of conjugation patterns in classical 形容詞 and 動詞, origin?
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/?"
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answer