Modify verilog CORDIC code for -360 to 360 degrees
You want to use the identities

$$\sin -\theta = -\sin \theta$$
$$\cos -\theta = \cos \theta$$
$$\sin(\theta + 90^\circ)=\cos \theta$$
$$\cos(\theta + 90^\circ)=-\sin \theta$$
$$\sin(\theta + 180^\circ)=-\sin \theta$$
$$\cos(\theta + 180^\circ)=-\cos \theta$$

From the comments in the code you linked: 

This code is for the first quadrant, but is easily extended to the full
circle by first doing a coarse rotation.  For example, to compute the
arctan of -y/x, in the second quadrant, feed the cordic function y/x and
then add 90 degrees (or pi/2 if using radian mode) to the result.  When
computing sin and cos of an angle, coarse rotate the angle into the first quadrant
by subtracting the appropriate number of 90 (or pi/2) increments to get the angle in
the first quadrant, keep track of this value, feed the cordic the angle.  Then
simply change the sign of the results based on this stored number.


In digital logic, the lowest-resource way to do the "coarse rotation" is probably to simply subtract 90 degrees iteratively, keeping track of the number of subtractions, until you get an angle less than 90. If you need to do it in a single cycle, you'll need several subtractors to do those calculations in parallel.