reference for 1962 IWC

Maybe
someone can help me, I have a IWC caliber 853 that I would like to know
the reference. Movement number 1646757, case numer 1563743. I think 1962.

Your information is correct:

ie Cal 853 from 1962, according to those serial numbers. There is nothing
further we can tell you without seeing photographs of the watch.

Is there no reference number engraved in the case back?

Now with photos

no number on case back

This should be the Reference number

No Number. Photo not great, it says...

Stay brite steel

International watch company

Swiss

Sorry

looked like numbers to me

Ref 666 with a re-done dial, though the dial may be redone and completely
unassociated.