  Generalized Toeplitz plus Hankel operators $T(a)+H_{\alpha}(b)$ generated by
functions $a,b$ and a linear fractional Carleman shift $\alpha$ changing the
orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces
$H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$
and satisfy the condition
  $$ a(t) a(\alpha(t))=b(t) b(\alpha(t)),\quad t\in \mathbb{T}, $$ the defect
numbers of the operators $T(a)+H_{\alpha}(b)$ are established and an explicit
description of the structure of the kernels and cokernels of the operators
mentioned is given.
