  The celebrated Touchard congruence states that $B_{n+p}\=B_n+B_{n+1}$ modulo
$p$, where $p$ is a prime number and $B_n$ denotes the Bell number. In this
paper we study divisibility properties of $B_{n-p}$ and their generalizations
involving higher powers of $p$ as well as the $r$-Bell numbers. In particular,
we show a closely relation of the considered problem to the Sun-Zagier
congruence, which is additionally improved by deriving \mbox{a new} relation
between $r$-Bell and derangement numbers. Finally, we conclude some results on
the period of the Bell numbers modulo $p$.
