The pinon, on the other hand, is symmetrical along its tooth length. Would it be possible to design a face gear that would be also symmetrical (yet not cause undue tooth wear) to allow the pinion to slide between outer and inner face gears, to enable the pinion to consecutively engage two concentric face gears?
tl;dr Not exactly.
Your face gear tooth space and pitch would need to match your pinion tooth, which is a constant thickness. The problem is you need that constant space width at multiple radii on the face gear. But if your pitch is fine enough on a large enough radius on the face gear that difference gets smaller. You could use "straight" face gear teeth but that would result in slightly tapered tooth spaces going from the OD to center. You would need to negatively profile shift the face gear teeth to ensure the pinion fits both spaces.
A crown on the pinion teeth should be used to avoid edge loading the side closest to the face gear center, which would concentrate your stresses and accelerate wear. But then it would be asymmetric.