Does an equivalent transformation of a signal to a spectrogram image exist in which the phase information is part of the resulting image?
If you create a volumetric (unflattened 3D) image, you can use 2 layers in the 3rd dimension to represent magnitude and phase, or real and imaginary components of a complex spectrogram output.

In a 2D color spectrogram, you could try using an orthogonal color mapping, for instance, in RGB space, red + green for magnitude and blue (or delta blue) for phase, or similar for real and imaginary components, or use YUV or HSL color spaces, etc.