How to Swap XY Axes for One ImagePlot in a Multi-Plot Graph?

If both images are plotted on the same axis, then no you can't do what you want. However, you could plot them onto separate axes to get the effect your looking for. For example, if your first image is on the Top and Left axes, you could append the second image using

Then you can do an axis swap on the axis that your image of interest is associated with.

Or you could work with layouts to create multi-image plots instead of having all images in one graph. Then you could swap axes for each graph freely. This has some benefits and drawbacks of course. For example, it is a bit more difficult to align axes with this approach.

If the image plots are in their own subwindow, whether inside a graph, panel, or layout, swapping those axes is trivial.