Vibration diagnostics based on vibroacoustic signal data belong to the most common ways to monitor the technical condition of equipment and technical structures. The paper considers the general issues of vibration-based diagnostics and shows that in general, it is required to monitor both axial and torsional oscillations, as well as the inclination angle, occurring during the operation of various technical objects. To comprehensively monitor these parameters, a hybrid vibration sensor is proposed, simultaneously implementing three operating modes: recording linear displacements of the vibrating object; recording the rotation angle of the object at its torsional oscillations; recording the object angular deviation from the vertical component of the natural local geomagnetic field, i.e., the inclinometer mode. The proposed hybrid sensor design is described, and a theoretical analysis of the sensor’s operation in each of the aforementioned operating modes is performed. The authors show that in the inclinometer mode the sensor actually operates as a fluxgate meter. Generalizing the results of the sensor’s operation simultaneously in all three operating modes, an equation for the total output data signal has been obtained, which allows for obtaining the required information on the current values of linear displacements and rotation and inclination angles by selectively filtering it with respective three filters tuned to specific frequencies. The experimental studies of the proposed hybrid vibration sensor confirmed its ability to record various vibrational disturbances and changes in the inclination angle of the monitored object.