The frequency shift of a helical light beam experiencing the rotation near the axis deferring from its own axis (conical evolution) is studied theoretically. Both the energy and the kinematic approaches lead to a paradoxical conclusion that after a whole cycle of the system rotation the beam does not return to its initial state. Another paradox is manifested in the peculiar behavior of the beam transverse pattern rotation at different geometric parameters of the evolving system. A fundamental role of the detecting system motion is substantiated. The special natural observers motion is found for which both paradoxes are eliminated. Relations of the described facts with the Hannays geometric phase concept are discussed.
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