In English, we have blink and wink where wink is a voluntary action. In German, which one is the voluntary one and which one is not? When should I use one over another? Or are both of them is a synonym?
The words blinzeln and zwinkern tell nothing about the number of eyes involved and careful investigation proves volition is no absolute criterion either!
On the number of eyes involved
The conjectures voiced in the other answer about the number of eyes involved in anblinzeln are completely misguided. You can definitely say:
Aus seinem verbliebenen Auge blinzelte mich der Pirat verständnislos an.
When I entered blinzeln m, my browser already proposed blinzeln mit einem Auge. It is used here and an doesn’t change anything about the number of eyes involved. An can be used to emphasise the voluntary nature but can also be directional.
I was initially inclined to agree with the claim in the comments that zwinkern involves only one eye, but more research shows this is not strictly required. Not only are there numerous counter examples counter example, second counter example, third counter example. The Duden also lists:
nervös, vielsagend, vertraulich [mit den Augen] zwinkern
To put the nail in the coffin, Grimm also notes:
allgemein gebräuchlich und verbreitet die engere bedeutung 'mit den augen zwinkern, mit den augenlidern zucken, sie zusammendrücken'
Grimm also mentions usages I never heard in the 20th century:
herr Barbarossa zwinkerte mit dem munde
It is my experience as a native speaker that zwinkern is usually voluntary and blinzeln involuntary. Blinzeln typically occurs when we wake up or when light blinds us. Thus it typically involves both eyes. But careful investigation reveals neither voluntary nature nor eye number strictly differentiates these words. Same holds for anblinzeln where an fulfils the role of English at. This can still be involuntary, and unintentional just as gape at/anglotzen or ankotzen/puke at or anstinken can. See below.
Indeed there are examples of blinzeln as a voluntary action in Grimm:
man biege nunmehr den kopf zurück und blinzle mit den augen. Göthe 52, 181;
Grimm also notes:
die augen halb öfnen und blinken lassen, oft um damit zu winken, aber auch unklar sehen, geblendet, halbblind sein
On the other hand there are examples of zwinkern as an involuntary action. One of Grimm’s definitions for zwinkern reads:
eine unbewuszte zuckende bewegung der augenlider
Response to comments: Anglotzen can be unintentional too Simply consider:
Er nahm mich nicht wahr, sondern glotzte mich aus toten Augen an.