Cocycles of \zm -actions on compact metric spaces provide a means for constructing m -actions or flows, called suspension flows. It is known that all m flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. In this paper we investigate when the space for a suspension flow depends only on the given \zm -action and not on the actual cocycle. The identity map $I$ of m determines perhaps the simplest cocycle for any \zm -action. We introduce invertible cocycles, and show that they produce the same space as the cocycle determined by the identity map $I$. The main result, Theorem 5.2, establishes an integration test for invertibility using piecewise linear maps and related topological ideas. Finally, it applies to the known methods for modeling m flows as suspensions and leads to refinements of these results.