The building block of in-memory computing with spintronic devices is mainly based on the magnetic tunnel junction with perpendicular interfacial anisotropy (p-MTJ). The resulting asymmetric write and readout operations impose challenges in downscaling and direct cascadability of p-MTJ devices. Here, we propose that a previously unimplemented symmetric write and readout mechanism can be realized in perpendicular-anisotropy spin-orbit (PASO) quantum materials based on Fe3GeTe2 and WTe2. We demonstrate that field-free and deterministic reversal of the perpendicular magnetization can be achieved using unconventional charge-to-z-spin conversion. The resulting magnetic state can be readily probed with its intrinsic inverse process, i.e., z-spin-to-charge conversion. Using the PASO quantum material as a fundamental building block, we implement the functionally complete set of logic-in-memory operations and a more complex nonvolatile half-adder logic function. Our work highlights the potential of PASO quantum materials for the development of scalable energy-efficient and ultrafast spintronic computing.