We present a new, to the best of our knowledge, type of vector three-dimensional dissipative optical solitons with more extended degrees of freedom in a laser or laser medium with saturable absorption. These solitons are reconfigurable, include polarization singularities, and have various mutual orientations of nearly toroidal localization domains of polarization components. Numerical modeling confirms the stability of these solitons and breathers and reveals their symmetry and even "supersymmetry," as well as transformations when parameters leave the stability region. These solitons, which have no scalar analogs, are capable of carrying more than one bit of information. Our results expand the "alphabet" of solitons and can provide a route to breakthroughs in larger-capacity communication and information applications.