For any shape, if you know the two moments, $I_{xx}$ and $I_{yy}$, there is a fascinating amount of study to arrive at the principal moments. You'd need to know $I_{xy}$ as well, which is harder to find tables on. Fortunately, an angle simply can be broken into two rectangles. Rectangles by definition have 0 for their product of inertia. So, the product of intertia for the angle is simply the area of the rectangles times the distance between the centroid of each rectangle in both x and y directions:

$$I_{xy} = A_1d_{x_1}d_{y_1} + A_2d_{x_2}d_{y_2}$$
Note that dimensions are, unlike we're used to, relative vectors - as such for the first rectangle, $d_{y_1}$ is positive, but $d_{x_1}$ is negative. For rectangle 2, the reverse is true.
With this, then the principal moments are simply:
$$\frac{I_{xx}+I_{yy} \pm \sqrt{I_{xx}^2+I_{yy}^2+4I_{xy}^2-2I_{xx}I_{yy}}}{2}$$
The angle can be found as
$$\theta = \frac{1}{2}\arctan{\frac{2I_{xy}}{I_{yy}-I_{xx}}}$$