In singular value decomposition, is there a difference between starting with transpose(A) * A or A * transpose(A)?

Question

When calculating the SVD for a matrix I go through these steps :

1/ Calculate tranpose(A)*A

2/ Find the eigenvalues for that matrix and deduce $\Sigma$

3/ Find the eigenvectors and deduce $V$

4/ Deduce $U$ from the relation $AV=U \Sigma$

Now suppose I started with A*transpose(A), what does change in these steps? Will I be calculating the eigenvalues of $U$ instead of $V$?