I am going to give the standard answer, although there might be exceptions to this due to some implicit assumptions.
The standard answer is that Supersymmetry is unique in implying things about vacuum energy or relating particles of different spin, or even different chirality (other than discrete symmetries like C,P,T), because it is the only nontrivial extension of the Poincare group. All other symmetries must commute with the Poincare group, meaning that acting with the symmetry will not affect the space-time representation of the particle, it will keep its helicity and its momentum.
The argument is due to Haag, Lopuszanski, Sohnius, and it is really a minor extension of the Coleman Mandula theorem, which itself is an extension of O'Raifertaigh's earlier result. The main conclusion is that the only graded extension of Poincare symmetry is the super-Poincare group. There is no other extension of Poincare symmetry.
When the theory is conformal, the Poicare group is extended to the conformal group, and the super-Poincare group to the super-conformal group. It is widely believed that this is the only real nontrivial extension of the ordinary Coleman Mandula result.
The idea of Coleman and Mandula's argument is to look at the transformation of S-matrix particle states. If the symmetry acts trivially on all the incoming particle states, they feel free to conclude that the full symmetry is a product of the Poicare symmetry and the internal symmetry.
If the symmetry carries a space-time index $\mu$ that is not a spinor, then the particles which carry this conserved quantity will have it be either constant as a function of the momentum (in which case the conserved quantity commutes with the Poincare group) or some polynomial combination the momentum with the right index structure. The reason is that the momentum of a massive particle is the only thing that changes from frame to frame, so a boost dependent conserved quantity will have to have a particular dependence on the momentum.
But then particle collisions will have to be constrained by the conservation not just of energy momentum, but by an additional constraint of the sum of some polynomial quantity in the momentum. Conservation of momentum, energy, and the new quantity leads the 2-2 particle scattering to only happen in certain special directions, and this is ridiculous--- it contradicts the local quantum field property, or even the more general analyticity of scattering as a function of angle.
This argument can be made as rigorous as you like. Historically, the main idea is originally due to O'Raifertaigh. Coleman noted that acting the symmetry operators again and again generates infinite dimensional families of particles in the SU(6) theory. Coleman and Mandula's theorem closed the subject. The whole history of this fascinating little sub-plot is summarized comprehensively in Dyson's lecture-note and reprint volume "Symmetry groups in Nuclear and Particle Physics".
Haag Lopuszanski Sohnius
From Coleman/Mandula, you can conclude that the only generators which act nontrivially on particles are those with no space-time index, or those with a spinor index. It is relatively straightforward to reconstruct the SUSY algebra uniquely from the Coleman Mandula theorem, and this is done explicitly in the first chapter of Wess and Bagger.
The main implicit assumption is that a symmetry must act on particle states nontrivially. It is completely possible that a symmetry is form-like in that it acts only on extended sheets, and becomes trivial when the sheets close on themselves. I don't know if this is a real loophole or not.
You asked about mirror symmetry and discrete symmetries. Mirror symmetry is not a symmetry of the physics, it is a relation between the geometry of two different manifolds that is due to the physical relations from T-dualizing cycles on the manifold in string theory. T-duality is not a symmetry either, it relates one formulation of a theory to a different formulation, and usually the word "symmetry" is reserved for things that relate two physically states in the same formulation. It can be thought of as a discrete global gauge symmetry of ground states of M-theory (meaning a global discrete identification of states, not a relation between different states), since it is identifying superficially different vacua as really the same thing. Mirror symmetry and supersymmetry "commute", in the sense that the mirror image of a particle and it's superpartners are the same particles and the same superpartners, it's the same stuff being described.
For discrete symmetries, N=2 theories often preserve parity, because they have supercharges of both helicities. R symmetry is a discrete symmetry that does not commute with the supersymmetry, and the Witten fermionicity index (-)^F is just preserved by general principles of quantum mechanics. There are other discrete symmetries in different models. I added this to answer the question you asked more completely (I was only addressing extending supersymmetry above), so as to prevent downvotes.