Having searched the term "involutorial" here, on Google, and in the references of the source paper where I first have met this term (in fact the only time), I am wondering whether could anyone give a definition of an involutorial homeomorphism. The location of this term in the paper is Theorem 2.4 on page 98.
Could it be that the source paper is dated 1955?
An involutorial homeomorphism is a homeomorphism that is also an involution. An involution is a function $f$ that satisfies $f^2=\operatorname{id}$, i.e. an function that is its own inverse. See also the Wikipedia page on involutions, which states at the outset that an involutory function is the same as an involution.