In the context of para-Kenmotsu geometry, properties of the $\varphi$-conjugate connections of some canonical linear connections (Levi-Civita, Schouten-van Kampen, Golab and Zamkovoy canonical paracontact connections) are established, underlining the relations between them and between their structure and virtual tensors. The case of projectively and dual-projectively equivalent connections is also treated. In particular, it is proved that the structure tensor is invariant under dual-projective transformations.