Generalized global symmetries are present in theories of particle physics, and understanding their structure can give insight into these theories and UV completions thereof.  We will particularly discuss the use of non-invertible chiral symmetries in BSM model-building. The identification of a non-invertible symmetry in Z' models of L_µ - L_τ reveals the existence of non-Abelian horizontal gauge theories which naturally produce exponentially small Dirac neutrino masses. Next we will uncover a subtler non-invertible symmetry in horizontal gauge theories of the quark sector which will lead us to a massless down-type quarks solution to strong CP in color-flavor unification. Intriguingly, this symmetry is present by virtue of the SM having the same numbers of colors and generations.