A new family of local hybrid functionals with a position-dependent mixture of local and exact exchange has been constructed. On the basis of conceptual similarities between local hybrids and explicit nondynamical correlation functionals, the relative spin polarization has been introduced as an additional variable into the local mixing function (LMF) that determines the position dependence of the exact exchange admixture. This leads to two new two-parameter LMFs, one based on the ratio of the local kinetic energy density to the von Weizsacker kinetic energy density and the other on the dimensionless density gradient. After optimization of the two free parameters for small test sets, significant improvements of atomization energies of the full G3 set are obtained (mean absolute errors of 2.96 and 3.18 kcal mol(-1), respectively), with minor deterioration for classical reaction barriers (HTBH38 and NHTBH38 test sets). The simultaneous description of two-center three-electron dimer cations remains a challenge. It is shown that the exact and local spin density exchange energy densities have closely related gauge definitions, so that gauge mismatch is only a minor problem for local hybrids based on local exchange.