Just as electrons exist in a "cloud", whole atoms and molecules do as well. The classic double-slit interference experiment has been performed with deuterium atoms and even molecules massing more than 25 kDa, such as tetraphenylporphyrins with fluoroalkylsulfanyl chains.
Then given that position and velocity of a particular hydrogen cannot be determined precisely, they could be interchanging continually -- though calculating the probability of such interchange is decidedly (no pun intended) difficult.
As a similar phenomenon, I find amazing the simple, inexpensive, Esaki or tunnel diode, in which electrons cross a barrier (well, disappear on on side and appear on the other) by quantum tunneling. Is that not happening to the hydrogens in different positions in ice?