  We present a novel class of calculable four dimensional composite
pseudo-Goldstone boson Higgs models based on symmetric G/H coset spaces which
contain a Higgs-parity operator V as well as a linear representation $\Sigma'$
for the Goldstone bosons. For such cosets the low-energy effective Lagrangian
for the Standard Model fields can have an enhanced global symmetry which we
call the maximal symmetry. We show that such a maximally symmetric case leads
to a finite and fully calculable Higgs potential, which also minimizes the
tuning by eliminating double tuning and reducing the Higgs mass. We present a
detailed analysis of the Maximally Symmetric SO(5)/SO(4) model, and comment on
its observational consequences.
