Is this possible, for a signal to be both left- and right-hand polarized?
Yes, it's very much possible:
While the superposition of two orthogonal circular polarizations might¹ indeed look linear (just as the superposition of a horizontal and a vertical polarized wave at appropriate phasing is a circularly polarized wave), of course that means that a linearly polarized wave is at the same time circularly polarized in both directions.
Technically, this is widely exploited: Satellite receivers use polarization multiplex. That is awesome, because you get two totally independently useful "subchannels", as long all media the wave travels through is a largely a linear medium and anisotropicisotropic. (And the microwave frequencies geostationary satellite downlink channel fulfills that pretty well.)
Even if that's not the case, you still get some isolation between RHCP and LHCP, and can use that for MIMO techniques to increase your data rate or robustness beyond what you can do on a single polarization.
¹ might because that's not necessarily the case. Remember the Poincaré sphere:
When you add waves of different polarizations, you wander on the surface of that sphere; only when you add RHCP and LHCP with the same magnitude, you end up with a linear polarization. The angle of that is then defined by the phase between the two constituent waves; every other combination, every attenuation that affects one rotational sense more than the other, will produce an elliptic polarization.
Let me rephrase your question:
Is this possible, for a wave to be both left- and right-hand polarized at the same time?
No, that's not possible, because any wave can only occupy one point in polarization space.