  We develop a complete Fredholm and invertibility theory for Toeplitz+Hankel
operators $T(a)+H(b)$ on the Hardy space $H^p$, $1<p<\infty$, with piecewise
continuous functions $a,b$ defined on the unit circle which are subject to the
condition $a(t)a(t^{-1})=b(t)b(t^{-1})$, $|t|=1$. In particular, in the case of
Fredholmness, formulas for the defect numbers are established. The results are
applied to several important examples.
