I finished Core 10000 (an Anki deck) and 振る舞う (to entertain) was one of the
last ones to pop up.

I find it interesting because to me it appears to be a combination of plain
form 振る (to shake) + another plain form verb 舞う (to dance). This differs from
all the other two-verb compounds I am used to such as 取り引き (transaction, noun
in this case)、書き直す (rewrite)、やり直す (redo)、吹き出す (blow out), 話し出す (start
talking)、飛び上がる (fly up), where the first verb is the masu-form instead of
plain.

I cannot recall any similar plain-form + plain-form verbs like 振る舞う.

My theory is that either this verb is an exception to the rule or this verb is
in fact, NOT a two-verb compound. It is simply ふるまう, a single verb, and the
Kanji was conveniently added later to make it look like a two-Kanji compound.
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verbs
Two-verb compounds: 振る舞う: plain-form + plain-form verb instead of masu-form + plain form?
You are correct, linking two verb in their 連体形 ("dictionary") form is unusual
and unexpected.

Both of the dictionaries 日本国語大辞典 and 広辞苑 suggest the following explanation:

> 「観智院本名義抄」や「色葉字類抄」で「翔」を「ふるまふ」と読んでいることから「羽を存分に振って自由舞う」という「振るい舞う」が原義かとおもわれる。

That is, it is a shortening of the original form _huruimau_ , in the sense of
a bird flapping its wings and taking flight, fluttering about in the skies (=
"dancing", the Japanese _mau_ is used in this sense). _hurui_ is the 連用形
("masu-stem") of the verb _huruu_ , and this explanation would render the
compound regular.

However, both dictionaries imply that this is only a hypothesis, and a
dictionaries main purpose is not to list etymological research. It is just as
well that the kanji are simple _ateji_ and that _hurumau_ derives from some
different word(s), or that it is simply irregular. The earliest usage a quick
dictionary lookup yields is from about 800 years ago, in the 古今著聞集.
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