Presumably it's always in that form.

@@ -13 +13 @@
-<s_inf>usu. 手を~</s_inf>
+<s_inf>as 手を〜</s_inf>

振り返す	7178
手を振り返す	4782

I think it should stay with that suggested gloss. Good to differentiate from
similar-looking terms.

@@ -13,3 +13,2 @@
-<gloss>to relapse</gloss>
-<gloss>to reassert</gloss>
-<gloss>to review</gloss>
+<s_inf>usu. 手を~</s_inf>
+<gloss>to wave back</gloss>

振り返す	7178
ぶり返す	27396

https://hinative.com/ja/questions/14754606
振り返す と 繰り返す はどう違いますか？説明が難しい場合は、例文を教えて下さい。

振り返すとは言いません。振り返り、か、振り返る、が正しいです。
振り返り(noun),振り返る(verb)：review, look back
繰り返す：repeat


eij:
 手を振り返す wave back
にっこり笑って手を振り返す wave back smiling

Results for "を振り返して" on google are exclusively things like "手を振り返す男性心理" "「後でね」って手を振り返して" "手を振り返さないのは嫌われてますか？" "あ、手を振り返してくれてる" "手を振り返してほしい"

We already have ぶり返す as a separate entry. Nikk alone does have 振り返す as both ふりかえす and ぶりかえす. Daij koj have ぶりかえす as ぶり返す only and don't have ふりかえす as an entry at all. My Windows IME does not bring up "振り返す" for the input ぶりかえす at all, though it does for ふりかえす (2nd option, after 振替す).

ふりかえす with the meaning attested in nikk is likely archaic and the only current meaning of this word is "to wave back", which explains the high ngram count. I suggest this entry is deleted to avoid confusion, or changed to "to wave back" (though it's a little A+B)

I'm just writing a comment because I'm not sure, but shouldn't be the reading ぶりかえす instead? Both weblio and kotobank gives me that reading (actually kotobank redirects me to ぶり返す).
In addition weblio has a ふりかえす entry, but it should be something like "wave back" and "look back".
https://www.weblio.jp/content/振り返す
https://kotobank.jp/word/ぶり返す-621334

nikk

@@ -12,0 +12,1 @@
+<pos>&vt;</pos>