Real is dual to imaginary -- complex numbers are dual. The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

And for higher order tensors like M_(ijkl), it is possible to have both symmetric and antisymmetric index pairs... right?

Right, 1/2 is pronounced 'one half' or 'a half'