We introduce an extension of the periodogram concept to time-resolved spectroscopy. USuRPER -- Unit Sphere Representation PERiodogram -- is a novel technique which opens new horizons in the analysis of astronomical spectra. It can be used to detect a wide range of periodic variability of the spectrum shape. Essentially, the technique is based on representing spectra as unit vectors in a multidimensional hyperspace, hence its name. It is an extension of the phase-distance correlation (PDC) periodogram we had introduced in previous papers, to very high-dimensional data like spectra. USuRPER takes into account the overall shape of the spectrum, sparing the need to reduce it into a single quantity like radial velocity or temperature. Through simulations we demonstrate its performance in various types of spectroscopic variability -- single-lined and double-lined spectroscopic binary stars and pulsating stars. We also show its performance on actual data of a rapidly oscillating Ap (roAp) star. USuRPER is a new tool to explore large time-resolved spectroscopic databases, e.g. APOGEE, LAMOST and the RVS spectra of Gaia. We have made available to the community a public GitHub repository with a Python implementation of USuRPER, to experiment with it and apply it to a wide range of spectroscopic time series.