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A narrow connection between infinite binary words rich in classicalpalindromes and infinite binary words rich simultaneously in palindromes andpseudopalindromes (the so-called $H$-rich words) is demonstrated.  The correspondence between rich and $H$-rich words is based on the operation$S$ acting over words over the alphabet $\{0,1\}$ and defined by$S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$.  The operation $S$ enables us to construct a new class of rich words and a newclass of $H$-rich words.  Finally, the operation $S$ is considered on the multiliteral alphabet$\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As abyproduct, new binary rich and $H$-rich words are obtained by application of$S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.

Constructions of words rich in palindromes and pseudopalindromes