Polarized neutron reflectometry (PNR) has been used to investigate the magnetic interlayer coupling in a MBE-grown Fe/Si/Fe(001) sandwich at room temperature and at 10 K. Both the magnitude and orientation of the magnetic moments of the Fe layers are obtained from a rigorous analysis of the PNR data. Orthogonal configurations of the Fe magnetizations were observed, providing unambiguous evidence for the presence of a biquadratic term in the exchange coupling energy. The competition between the bilinear and biquadratic exchange couplings results in distinct orthogonal and antiparallel configurations of the Fe magnetizations at room temperature. A previously unresolved magnetic configuration in the room-temperature hysteresis curve was identified by the PNR measurements as a (formula presented) spin-flop transition. The dominant role of the biquadratic coupling at low temperatures is evident from the orthogonal configuration of the magnetizations at remanence in the measurements at (formula presented) The magnetic configurations deduced by PNR are in good agreement with those obtained by fitting the magnetic hysteresis loops using a global energy minimum calculation.