An interval of exceptional solar activity was registered in early September 2017, late in the decay phase of solar cycle 24, involving the complex Active Region (AR) 12673 as it rotated across the western hemisphere with respect to the Earth. A large number of eruptions occurred between 4-10 September, including four associated with X-class flares. The X9.3 flare on 6 September and the X8.2 flare on 10 September are currently the two largest during cycle 24. Both were accompanied by fast coronal mass ejections (CMEs) and gave rise to solar energetic particle (SEP) events measured by near-Earth spacecraft. In particular, the partially-occulted solar event on 10 September triggered a ground level enhancement (GLE), the second GLE of cycle 24. A further, much less energetic SEP event was recorded on 4 September. In this work we analyze observations by ACE and GOES, estimating the SEP event-integrated spectra above 300 keV and carrying out a detailed study of the spectral shape temporal evolution. Derived spectra are characterized by a low-energy break at few/tens of MeV; the 10 September event spectrum, extending up to $\sim$1 GeV, exhibits an additional high-energy rollover at several hundred MeV. We discuss the spectral interpretation in the scenario of shock acceleration and in terms of other important external influences related to interplanetary transport and magnetic connectivity, taking advantage of multi-point observations from STEREO. Spectral results are also compared with those obtained for the 17 May 2012 GLE event, associated with the previous GLE.