Identifying a pseudovector

The very fact that you need a "hand rule" shows that the magnetic field is a pseudo-vector. While electric current is a true vector, the magnetic field is actually the space part of the EM field tensor, which is a rank 2 antisymmetric tensor. If you write out the EM field tensor, you will see that the E field is the time-space part of it, and the B field is the space-space part of it. Since the B-field includes two space indices, it does not change sign when you change the sign of the space coordinates, while the E-field, which contains only one space coordinate, does change sign. So under spatial transformations, the E field behaves as an ordinary vector, while the B field behaves as a pseudo-vector. Under Lorentz transformations, the E and B fields mix, so neither one transforms as a vector and you need to use the complete EM field tensor.