Calculate determinant of unitary matrices based on SVD implementation

I have a real square matrix X which I need to perform a Singular Value Decomposition on. Now, performing the operation

X=USV^T

as U and V are orthogonal, we know that det(X)=±det(S) and det(S) is non-negative as singular values are non-negative.

Now, I need to know the sign of the determinants of U and V (which is the same as knowing the determinants, of course). However, the naive approach costs me 2 O(N^3)

I was wondering whether someone knows of a way to either

1. Infer the sign of the determinants of U and V as a bi-product of the SVD-implementation in numpy,scipy or a similar library in Python, without having to call det(U) and det(V).
2. Calculate the determinant of an orthogonal matrix which is faster than the default implementation of det(U), based on the fact that U/V is orthogonal.