I used PCA in Design of Experiment applications several times. It does work. You can find a DoE presentation for free download in my publications.

The PCA relates the output variables (called response in a DoE) and tells you about the true dimensionality of the output space.

The DoE delivers a structured plan for the input variables (called factors in the DoE).

Although the PCA is not a regular part of the DoE method, it is a nice and powerful combination.

Looking at your crossover design as a small DoE, it should work in the same way.

Karl Siebertz Thank you for your reply. In this case, does the independence of observations become an issue? Do the paired observations violate any assumptions of PCA?

If the goal is to understand the effect of the experimental treatment, then don't use PCA. It's an unsupervised technique that doesn't take your experimental design into account. I'd recommend this paper by Hervé et al., especially the section about redundancy analysis (RDA). I think something like RDA is probably what you're looking for. Hervé, M. R., Nicolè, F., & Lê Cao, K.-A. (2018). Multivariate Analysis of Multiple Datasets: a Practical Guide for Chemical Ecology. Journal of Chemical Ecology, 44(3), 215–234. https://doi.org/10.1007/s10886-018-0932-6

Eric Scott Thank you for your reply. I am not using PCA to investigate the effects of the experimental treatment. I am aware of the fact that it doesn't take into account the experimental design. My goal is to use PCA as a dimension reduction method of my data set.

Then I think you're free to do what you want with PCA. It doesn't really have any assumptions, per se, as it isn't testing anything. I think it seems reasonable to do what you're suggesting. One other option would be to use the difference between each factor level for each individual in a PCA, if that makes any sense for your data.

Eric Scott Thank you for your reply, I highly appreciate it.