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The verb ひるがえす feels like it is made of two part ひる and かえす but I could not
find any references backing this interpretation. And I can't figure out what
would be ひる.

@snailboat also mentioned くつがえす whose meaning is very close to ひるがえす. If an
answer could solve both cases it would be much appreciated.
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etymology compounds
Is 翻す a former compound verb that merged into its current form?
[翻]{ひるがえ}す is a transitive verb of [翻]{ひるがえ}る.  
It seems that it shows the movement of a flag originally.

This theory remains a matter of speculation.

> 翻る…[旗]{はた}が翻る(A flag flaps)  
>  →ひらひらと[裏]{うら}[返]{がえ}る→ひるがえる

others:

> [蘇]{よみがえ}る…[死者]{ししゃ}が蘇る(The dead revive)  
>  →[黄泉]{よみ}から[帰]{かえ}る→よみがえる

※黄泉 is the land of the dead.

I did not understand about [覆]{くつがえ}る...  
  
  
(about a hypothesis)  
This is an ancient Japanese poem([和歌]{わか})  
たみのなけきに むかつひめ いそききしいに ゆきひらき  
民の嘆きに 向津姫 急ぎ紀志伊に 行き翻らき [link](http://gejirin.com/gsrc/hi/hiraku.html)
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## 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](https://kotobank.jp/word/%E3%81%8F%E3%81%A3%E3%81%A4%E3%82%8A) and
[Weblio](http://www.weblio.jp/content/%E3%81%8F%E3%81%A3%E3%81%A4%E3%82%8A),
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](https://japanese.stackexchange.com/questions/18729/%E3%81%A3-%E3%82%8A-form-
adverbs/36660#36660).)

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.
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