  Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is
known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear
form $b$ if and only if $V$ is self-dual. We show that whenever a Morita
bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and
$B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence
preserves self-duality. Even more, if the bilinear form on $M$ is symmetric
then for $p$ odd the correspondence preserves the geometric type of simple
modules. In characteristic 2 this holds also true for projective modules.
