  For most classical and similitude groups, we show that each element can be
written as a product of two transformations that a) preserve or almost preserve
the underlying form and b) whose squares are certain scalar maps. This
generalizes work of Wonenburger and Vinroot. As an application, we re-prove and
slightly extend a well-known result of M{\oe}glin, Vign\'{e}ras and Waldspurger
on the existence of automorphisms of $p$-adic classical groups that take each
irreducible smooth representations to its dual.
