Mainstream deep learning-based dysarthric speech detection approaches typically rely on processing the magnitude spectrum of the short-time Fourier transform of input signals, while ignoring the phase spectrum. Although considerable insight about the structure of a signal can be obtained from the magnitude spectrum, the phase spectrum also contains inherent structures which are not immediately apparent due to phase discontinuity. To reveal meaningful phase structures, alternative phase representations such as the modified group delay (MGD) spectrum and the instantaneous frequency (IF) spectrum have been investigated in several applications. The objective of this paper is to investigate the applicability of the unprocessed phase, MGD, and IF spectra for dysarthric speech detection. Experimental results show that dysarthric cues are present in all considered phase representations. Further, it is shown that using phase representations as complementary features to the magnitude spectrum is very beneficial for deep learning-based dysarthric speech detection, with the combination of magnitude and IF spectra yielding a very high performance. The presented results should raise awareness in the research community about the potential of the phase spectrum for dysarthric speech detection and motivate further research into novel architectures that optimally exploit magnitude and phase information.