It's easy to construct mappings $R,S:A\to A$ of an annulus $A$ on the plane such that $R$ is a rigid rotation, $S$ possesses the intersection property, but $R\circ S$ does not. However, I'd like to have a reference to such an example.
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Here is a relevant reference:
M.B.Sevryuk. On the composition of mappings possessing the intersection property. Nauka, Tekhnika i Obrazovanie (Science, Technology, and Education), 2018, N 4(45), p. 6-9.
See https://3minut.ru/images/PDF/2018/45/NTO-4-45.pdf
This is a Russian journal but Sevryuk's note is in English.
Here is a reference with a modified example:
M.B.Sevryuk. Three examples in the dynamical systems theory. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 2022, v. 18, 084 (13 pages).
See https://doi.org/10.3842/SIGMA.2022.084 as well as https://arxiv.org/abs/2209.02620.