Wiener-Hopf plus Hankel operators W (a) + H(b) : L p (R + ) → L p (R + ) with generating functions a and b from a subalgebra of L ∞ (R) containing almost periodic functions and Fourier images of L 1 (R)-functions are studied. For a and b satisfying the so-called matching condition a(t)a(−t) = b(t)b(−t), t ∈ R, we single out some classes of operators W (a) + H(b) which are subject to the Coburn-Simonenko theorem. 2010 Mathematics Subject Classification: Primary 47B35, 47B38; Secondary 47B33, 45E10.