A machine (M) for mounting and demounting a tyre (P) relative to a corresponding rim (C) of a vehicle wheel (R) comprises a wheel-holder unit (200) that is movable in rotation about a vertical axis of rotation (Y) and movable in translation along an axis of translation (X) towards and away from the machine (M). The machine (M) comprises a main arm (400) and a working unit (500) connected to the main arm (400) and including a first arm (501a) pivoted to the main arm (400) at a hinge point (F) and a first sidewall pressing tool (502a) which is configured to come into abutment against a sidewall of the tyre (P) of the wheel (R) mounted on the wheel-holder unit (200) and which is connected to the first arm (501a). The main arm (400) is movable by translation parallel to the horizontal axis of translation (X) towards and away from a column (101) to keep the hinge point (F) vertically aligned with the vertical axis of rotation (Y) of the wheel-holder unit (200), responsive to a movement in translation of the wheel-holder unit (200).