Proposed Manuscript Chronology

[npcompl33t]

How are those "indexes of liberalism vs conservatism" computed? Perhaps from ... the way those representatves vote?
Sure, for any data set and any axis, the data points "contain" a sequential ordering along that axis.

Either there is a dominant axis or there is not. Anyone familiar with modern politics knows that it is bipolar and regardless of how people vote, most could identify based on policy positions where a representative would fall on the spectrum. Every data set does not contain a dominant sequential ordering, that is core to this entire debate and the literature I shared on Horseshoes in particular. 

I'm not claiming all PCA always produces meaningful gradients, or that PCA dimensions correspond to real world traits as opposed to abstract mathematical relationships. I agree with you on all these points.

Where we disagree appears to be your assertion that a horseshoe does not contain a dominant ordering that directly ties to a single linear feature like time or temperature.  I shared multiple studies and even textbooks where people identify discuss this at length, and recover and identify the dominant sequential ordering from horseshoes in a number of real-world contexts. I feel extremely confident my findings here are legitimate, and supported by the broader research.