X-ray free-electron lasers (FELs) are cutting-edge research instruments employed in multiple scientific fields capable of analyzing matter with unprecedented time and spatial resolutions. Time-resolved measurements of electron and photon beams are essential in X-ray FELs. Radiofrequency (RF) transverse deflecting structures (TDSs) with a fixed streaking direction are standard diagnostics to measure the temporal properties of the electron beams. If placed after the undulator of the FEL facility, TDSs can also be employed to reconstruct the power profile of the FEL pulses. We present measurements of an X-band RF TDS system with variable polarization with a resolution below one femtosecond. We show FEL power profile measurements with associated root mean square pulse durations as short as 300 attoseconds. The measurements have been carried out at Athos, the soft X-ray beamline of SwissFEL. Measurements with variable polarization and attosecond resolution are essential to characterize and optimize the electron beams in all its dimensions for all types of X-ray FEL experiments, in particular for ultrafast X-ray applications.