  The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on
the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$
belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition
  \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R},
  \end{equation*} we establish necessary and also sufficient conditions for the
operators $W(a)+H(b)$ to be one-sided invertible, invertible or generalized
invertible. Besides, efficient representations for the corresponding inverses
are given.
