  The paper is focused on the study of continuous orbit equivalence for
generalized odometers (profinite actions). We show that two generalized
odometers are continuously orbit equivalent if and only if the acting groups
have finite index subgroups (having the same index) whose actions are piecewise
conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally
established for $\mathbb Z$-actions. As a corollary we obtain a dynamical
classification of the restricted isomorphism between generalized Bunce-Deddens
$C^*$-algebras. We also show that the full group associated with a generalized
odometer is amenable if and only if the acting group is amenable.
