There are a variety of symmetries in standard theories of subatomic particles. For instance, the fact that the three colors of quarks are identical can be considered a symmetry.
Charge conjugation is the symmetry of their being an identical antiparticle for every particle.
If protons exist, an antiproton is expected to exist with exactly the same mass and other properties, only with opposite quantum numbers. Parity is the symmetry that reflects the fact that reflecting the universe about a point or a plane should give a universe of equal physical validity. When we studied magnetic fields, the direction of the magnetic field depended on our choice of a right-handed convention. However, magnetic fields only manifest themselves through the forces felt by the charged particles through the magnetic field. The calculation of the force also hinged on a right-hand rule.