Disambiguation of Anode and Cathode Terms in Diodes
A silicon diode (of any type) is basically just two doped regions of silicon (with the exception of PIN diodes which have an intrinsic region).



Source: Wikibooks

The Anode and cathode are defined by the dopants (group III for P and group V for N). All diodes exhibit avalanche behaviour in reverse; a zener is simply defined for where reverse conduction occurs.

As a note, most zeners below about 6V operate in true zener effect; above this they are generally avalanche effect devices.

As noted, the anode and cathode remain that way even in reverse bias.