Does the motor make a noise and try to turn, but with no power when reverse switch is thrown? If so, then the start winding is disengaged when you flip the reverse switch.
As to why the SSR doesn't work the other way around, I am not sure, but you could test this by simply moving the leads from one side of the SSR to the other. If it reverse starts but doesn't forward start, then you know it is the SSR causing the problem (which it appears to be true, from the information that you gave).
If it is the SSR, there is a solution to this, which is fairly simple. You can PM me if you find the above to be true and want to know the simple solution.