Relationship between O(n)- and SO(n)-representations?

Question

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal $O(n)$-representations that are equivalent as $SO(n)$-representations. Is there a nice (computable) way of telling whether $V$ and $W$ are equivalent as $O(n)$-representations?