Does the velocity vector always point in the same direction as the momentum vector? I was told that the angular velocity vector does not always have to point in the same direction as the angular momentum vector.  This is due to the fact that they are related by the equation $L=I \omega$.  But in general, $I$ is a tensor and so the result might not be in the same direction as $\omega$.  Because the equations for linear and angular motion are very symmetrical that leads me to ask -- does the velocity vector always point in the same direction as the momentum vector?
 A: 
Because the equations for linear and angular motion are very
  symmetrical

In Newtonian mechanics, linear momentum is a vector while angular momentum is pseudo-vector which hints at its true nature as a higher rank tensor object.
In relativistic mechanics, four-momentum is a four-vector while angular momentum is a (rank 2) four-tensor.
So, the 'symmetry' isn't really there.  Linear momentum and angular momentum are different kinds of objects.


does the velocity vector always point in the same direction as the
  momentum vector

As far as I know, linear kinetic momentum and linear velocity are parallel in both classical (three-vector) mechanics
$$\mathbf p = m \mathbf v$$
and relativistic (four-vector) mechanics.
$$\mathbf P = m \mathbf U $$
However, as yuggib points out in a comment, the canonical momentum
$$\mathbf P = \frac{\partial \mathcal L}{\partial \dot {\mathbf q}}$$
is not generally parallel to $\dot {\mathbf q}$.  For example, the canonical momentum of a non-relativistic charged particle is
$$\mathbf P = m\mathbf v + q\mathbf A = \mathbf p + q\mathbf A$$
where $\mathbf A$ is the magnetic vector potential.
