  A criterion given by Castejon-Amenedo and MacCallum (1990) for the existence
of (locally) hypersurface-orthogonal generators of an orthogonally-transitive
two-parameter Abelian group of motions (a $G_2I$) in spacetime is re-expressed
as a test for linear dependence with constant coefficients between the three
components of the metric in the orbits in canonical coordinates. In general, it
is shown that such a relation implies that the metric is locally diagonalizable
in canonical coordinates, or has a null Killing vector, or can locally be
written in a generalized form of the `windmill' solutions characterized by
McIntosh. If the orbits of the $G_2I$ have cylindrical or toroidal topology and
a periodic coordinate is used, these metric forms cannot in general be realized
globally as they would conflict with the topological identification. The
geometry then has additional essential parameters, which specify the
topological identification. The physical significance of these parameters is
shown by their appearance in global holonomy and by examples of exterior
solutions where they have been related to characteristics of physical sources.
These results lead to some remarks about the definition of cylindrical
symmetry.
