The signature of a zone is ‘one up from’ the signature of each of its facets.The siganture <+ + + − −> is coded here as (3, 2). Signature (m, n) is one up from (m', n') iff (m, n) = (m'+1, n') or (m, n) = (m', n'+1).
Since the signature of the metric of a facet is determined by the edge lengths thereof, two neighbors will agree on the signature of their shared facet. Whether the two neighbors can agree on a affine transport across their shared facet depends on whether their own signatures agree, which I claim is just whether their (n-dimensional content)2, as defined by the determinant of the metric tensor, has the same sign. As a program logic principle we require each zone to have the same sign of the determinant of the metric tensor.
Here is some confusion to avoid. Two neighbors assign the opposite sign to the Grassmann magnitude of their shared facet. We speak above, however, about the sign of the square of the Grassmann magnitude, which is a real. The topological orientation of the simplexes, as determined by the parity of their vertex lists, does not control the signature of the resulting metric—only the specified squared edge lengths do this.