Chiral materials lacking mirror symmetry can exhibit unconventional spin-orbit fields, including fully momentum-aligned radial Rashba fields as seen in twisted van der Waals homobilayers. We theoretically study Cooper-pair transfer in superconductor/ferromagnet/superconductor Josephson junctions with crossed (tangential and radial) interfacial Rashba fields. We find that their interplay leads to what we call the unconventional supercurrent diode effect (SDE), where supercurrent rectification occurs even with collinear (with respect to the current) barrier magnetization, not possible for conventional spin-orbit fields. This SDE, distinct from conventional Rashba-induced effects on Cooper-pair momenta, arises from the spin precession in the magnetic barrier. We propose it as a sensitive probe of chiral spin textures.