The following equivalence
relation is introduced: Two (Hausdorff) spaces X and Y are bijectvely related if there
exist continuous bijections f : X → Y and g : X → Y . This first paper considers the
case in which X and Y are connected manifolds. If either f or g is not a
homeomorphism, then each space is necessarily non-reversible and hence this
study produces more knowledge of such spaces. The chief results here are the
existence theorem (Theorem 2) and, perhaps, Corollary 12, which states that a
simply-connected manifold having only compact boundary components is
reversible.