Are conservative justices more likely to decide in favor of conservative plaintiffs than liberal plaintiffs, and are liberal justices more likely to find in favor of liberal plaintiffs? 
Paul Waldman pretty much lays it out about a new study that shows bias in a way even a conservative can’t ignore:
What jumps out is how different the conservative justices are from the liberal justices. While all the liberal justices ruled more often for liberal plaintiffs than conservative ones, the differences tended to be pretty small. But for the conservative justices, the differences ranged from large to enormous. While the headline on the times story reads, “In Justices’ Votes, Free Speech Often Means ‘Speech I Agree With,'” they could have titled it, “In Conservative Justices’ Votes, Free Speech Usually Means ‘Speech I Agree With,'” and it would have been more accurate.
We don’t know whether this pattern holds in other kinds of cases, but I wouldn’t be at all surprised if it did—if, for instance, the liberal justices were slightly more likely to strike down a conservative regulation, while the conservative justices were much more likely to strike down a liberal regulation. And this kind of pattern has analogues in many different areas of politics.
For instance, we all know that Washington has become more “polarized” in recent years, and if you listened to the way the media talk about it, you’d think that Democrats and Republicans have moved away from the center at similar speeds, each becoming equally opposed to compromise and intransigent. But that isn’t true. According to the Poole-Rosenthal DW-NOMINATE data, which are the state of the art in measuring congressional votes, what has happened in recent years is that Democrats in Congress have become slightly more liberal, while Republicans have become massively more conservative. (If you go to this page, you can see the data by scrolling down to the charts labeled “Party Means on Liberal-Conservative Dimension” for both the House and Senate.)
Or to take another area, both liberals and conservatives are sometimes confronted with cognitive dissonance when the world turns out not to be how they would like it to be. But while liberals end up being slightly more likely than conservatives to believe some things that aren’t true, conservatives are hugely more likely, on a range of topics from climate change to whether we found Saddam Hussein’s fictional weapons of mass destruction to whether Barack Obama is a United States citizen, to simply deny reality and convince themselves that the world accords with their preferences (Chris Mooney documented many of the ways conservatives are more likely to engage in “motivated reasoning,” processing information through the filter of their pre-existing beliefs, in his book The Republican Brain).
There are other things you could add to that list, depending on your opinions. Most Democrats I know would probably say that while people on their side sometimes play dirty in campaigns, Republicans always play dirty, and much dirtier. On the other hand, Republicans would probably say just the opposite. But this study provides some pretty compelling evidence that when you last heard Antonin Scalia say that all he cares about is the intent of the Founders, and you thought he was full of crap, you were absolutely right.
Conservatives are mongrels and liberals are unicorns. Conservatives want it all and they want it all now, while liberals will settle for their fair share. Conservatives want to win at all cost and liberals want to compete, but winning is secondary to the journey. That’s the way of the world. The funny thing is though that in the end the conservatives always come out second best. Because the masses will only be screwed over for so long before they rise up and clutch the 1% by their greedy, lying throats. That time is very near.