How much symmetry a theory is allowed to have

A helicity is a whole number or a whole number plus a half, of either sign. Supersymmetry gives a theory some number $N$ of operations, and each of the $N$ raises the helicity of the particle it acts on by $\tfrac12$. In reaching any one particle no operation can be used twice, and applying two of them in either order gives the same particle. So start from the particle of lowest helicity and reach the others by applying operations. Which particle you arrive at is fixed by which operations you used, and different sets give different particles. Since none repeats and the order does not matter, that is a set chosen from the $N$, and the particle's helicity is the lowest helicity plus half the number of operations in the set. The theory's content is the count of particles at each helicity. Take the theory to hold exactly one such collection; it is called a SupermultipletThe set of particles that the supercharges map into one another, treated as a single object.. Here is the whole of that worked through for $N = 2$, starting from a lowest helicity of $-\tfrac12$. Call the two operations $1$ and $2$:

  operations used     helicity      how many
  ---------------     --------      --------
  none                  -1/2            1
  {1}   or   {2}           0            2
  {1,2}                 +1/2            1

Four particles: one at $-\tfrac12$, two at $0$, one at $+\tfrac12$. The counts $1\ 2\ 1$ are the number of subsets of each size, and they add to $2^2 = 4$. Particles of whole-number helicity are bosonsParticles of whole-number helicity are bosons; those of half-integer helicity are fermions., and those of half-integer helicity are fermions. So this theory holds two bosons and two fermions. Note the span. Every operation raises helicity by $\tfrac12$, so using all $N$ of them raises it by $N/2$, and the top of the list therefore sits exactly $N/2$ above the bottom. For $N = 2$ that span is $1$, which is the distance from $-\tfrac12$ to $+\tfrac12$. Two more rules apply. Flipping the sign of every helicity in the list has to give back the same list, counts included, which is why the $N=2$ example was started at $-\tfrac12$ rather than anywhere else. And no consistent interacting theory contains a particle of helicity above $+2$ or below $-2$.