To reproduce:
Place boat in water push or spawn animal (horses will not do it) in to boat, jump in boat with animal row left or right then hit shift while holding A or D and the animal will carry on rowing round and round.
If both are pressed at the same time as jumping out then the boat will stay where it is but the animation for row will carry on.
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Confirmed
Video
Confirmed for 15w42a
Confirmed for 15w43a
Confirmed for 15w44a

Confirmed for 15w44b

Confirmed for 15w45a
Confirmed for 15w46a

Confirmed for 15w47a. The boat also keeps turning.

Confirmed for 15w47c

This can only be seen by the player who exited the boat.
Works on land & water.
Works with both player, mob and animal.

Confirmed for 15w51b

Still in 16w03a.

Cannot confirm completely fixed for 16w04a
The head of the mob starts rotating and once you enter the boat again the boat "finishes" rotating

Affects 16w05a.

Rotating mob head confirmed for 16w05b

Rotating mob head confirmed for 16w07a

Rotating mob head confirmed for 16w07b.

Confirmed rotating mob head for 1.9-pre1.
Also: The boat starts rotating again when the player gets in.
(And the name of this issue is now incorrect.)

New title suggestion?

All of what Steven said, confirmed for 1.9-pre2.
I can't help but think this is two disconnected issues. Maybe "When player leaves boat in motion, motion resumes when player returns to boat." and "Mob head rotation in boat is stuck."

@@unknown They are probably caused by the same bug.
@@unknown Passenger rotation keeps changing after getting out of a boat ?

Also, confirmed for 1.9-pre2

Affects 1.9.4.

Changing title per comments.

I've noticed that when you get back in, the boat continues its rotation. Might have something to do with it??
@unknown, that would be MC-96365.

Still an issue, three years later, in 1.13-pre5
Can confirm in 20w48a.
Can confirm in 21w03a.
Can confirm in 21w05b.

Can confirm in 1.18 pre-release 8
[media]
can confirm in 1.20 Pre-release 1.

Can confirm 1.20.4.