ひるがえす・ひるがえる
Nariuji's answer above for 翻す matches what I've researched in the past. ひる, ひら, even ふる (as in 振る) all seem to be related to ideas of something flat waving or flapping, cognate with 鰭{ひれ} fin, and possibly even cognate with 簸{ひ}る (ancient root verb ふ) to winnow (by shaking grain).
くつがえす・くつがえる
Looking specifically at 覆す・覆る, this appears to be a compound of くつ + かえす・かえる.
The かえす・かえる Part
One of the first senses listed for かえす, spelled using the kanji 反 or 返, is:
事物{ものごと}や事柄{ことがら}の位置{いち}を逆{ぎゃく}にする。
To reverse the positions of things or facts.
かえる is simply the intransitive side of this same meaning:
事物{ものごと}や事柄{ことがら}の位置{いち}が逆{ぎゃく}になる。
The positions of things or facts become reversed.
The くつ Part
But the くつ portion remains a bit mysterious.
One etymon (root word) for くつ that seems to make sense semantically here is 靴 shoes -- as in 靴反す to reverse the position of one's shoes → to make one's shoes on the top → to turn upside down.
As an alternative possibility, I note that くつ is also an old reading for 口, only used in very old compounds (such as 轡{くつわ} = 口 + 輪 = mouth ring = the bit for a horse). So perhaps くつがえす = 口反す to reverse the position of one's mouth → to make one's mouth on the bottom → to turn upside down.
However, both options here refer specifically to a person's body, whereas 覆す・覆る can be used for all kinds of things.
A more likely etymon is probably the くつ in less-used adverb くっつり sufficiently, adequately; completely. This word is missing from both Kotobank and Weblio, but I do find it in my dead-tree 国語大辞典 from Shogakukan. This may be the same くつ in くっつく to really stick to something (sometimes unsatisfyingly explained as 食う + 付く).
Many adverbs of the form CVCCV + り
, such as くっつり or がっくり or ゆったり, appear to be derived from gemination (double-consonanting) of the underlying two-mora root + り from あり. So ゆったり is from ゆた (with an underlying meaning of loose in terms of roomy, not constricting), observable in part by the synonymous adverb ゆたゆた, while がっくり is from がく (with an underlying meaning of loose in terms of able to wiggle about), with synonym がくがく. (See this related thread talking about such -り adverbs.)
So くっつり would appear to derive from くつ (with an underlying meaning of sufficiently; completely).
Putting It Together
From this, くつ + かえす = くつ反{がえ}す = to completely turn something over, which seems to best match the meaning of 覆{くつがえ}す. So too can we derive くつ + かえる = くつ反{がえ}る = to completely turn over (on its own), which seems to best match the meaning of 覆{くつがえ}る.