I came across the following table, which gives the percentage of the time that each pair of justices of the U.S. Supreme Court agreed with each other in non-unanimous decisions:
The table above, though, is sorted in a conservative-to-liberal order derived from the alignment data. As I originally saw the table, the justices were in the order Roberts, Stevens, Scalia, Kennedy, Souter, Thomas, Ginsburg, Breyer, Alito. This is the Chief Justice followed by the eight Associate Justices in the order in which they were appointed.
These nine people are basically just names to me -- and in fact, some of them aren't even that. I hadn't heard of some of these people until today. Anyway, here's how I did the sorting: Thomas and Stevens agree with each other least often, so one assumes they have the largest ideological difference; put them first and ninth in the order. Scalia is the justice who's most likely to agree with Thomas, so put him second; similarly Ginsburg is mostly likely to agree with Steens, so put her 8th. Roberts is most likely to agree with Scalia of the five justices who have yet to be picked, so put him third. Alito and Roberts are the two justices who are most likely to agree with each other, so clearly they should be next to each other in this ordering; put Alito fourth. Thus, we so far have
Thomas, Scalia, Roberts, Alito, ?, ?, ?, Ginsburg, Stevens.
(Note that the people on the left here are more conservative than the ones on the right, the reverse of what you'd expect from the usual use of "left" and "right" as "liberal" and "conservative".) We still have to position Kennedy, Souter, and Breyer. At this point I created the following table:
It's a tough call, but I figured that since Souter is the most likely to agree with Ginsburg, they should be next to each other; thus Souter gets the seventh slot. So now we have to put Kennedy and Breyer in order. Thomas, Scalia, Roberts, and Alito were all more likely to agree with Kennedy than with Breyer; Souter, Ginsburg, and Stevens were all more likely to agree with Breyer than with Kennedy. On this basis, Kennedy gets the fifth slot, Souter the 6th, which gives the order you can read off the first row of the big table which opened this post.
What surprised me is how coherent the assignment ended up being. If you look at the table above, for the most part each row or column increases smoothly to 100 and then decreases smoothly. (The coloring marks the exceptions; each pair of adjacent entries that's in the same color is a pair in the "wrong" order.) Furthermore, if you switch the order of any two adjacent justices you create more exceptions; this is clearly at least a local minimum. What this basically says is that the current court is one-dimensional; on most cases you can probably draw a line between two people in the ordering I gave and all the "yes" votes will be on one side, all the "no" votes on the other. (I may check this at some future date.)
I'm kind of curious if this could be done for, say, the Senate, although since the Senate has a hundred members and I'd have to compile the data myself I'm not going to do it. This took maybe ten or twenty minutes once I had the table, and was an amusing break from what I was working on this morning; doing it for the Senate would be actual work.
What's more, this ordering is in order with what the Supreme Court Institute has to say in the narrative parts of their report. Kennedy was in the majority in all 22 5-4 decisions the court made this term. They call Scalia a "conservative pole", though, which doesn't line up with my data.
Also, it would probably be possible to incorporate past justices into this ordering, if they'd overlapped with enough of the current justices that where they fell in the ordering was obvious. But I doubt that it's possible to rank all 110 justices this way; for one thing, our left-right distinction is different from the one that's existed at various points in history. And even if "left" and "right" had carried the same meaning over the last two hundred years, we wouldn't be able to compare every pair of justices -- just those that had overlapped -- and extending that sort of partial order to a total order is hard.