  An element of a group is \emph{reversible} if it is conjugate to its own
inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse
by an involution. A group element is strongly reversible if and only if it can
be expressed as a composite of two involutions. In this paper the reversible
maps, the strongly reversible maps, and those maps that can be expressed as a
composite of involutions are determined in certain groups of piecewise linear
homeomorphisms of the real line.
