The Taj Mahal is symmetric with respect to right-left reflection. That is to say, if you create a mirror image of it that flips right and left, you get the exact same thing. Therefore, if as we're standing outside looking at it my friend Jen tells me that she thinks the left side of the Taj Mahal is much nicer than the right side, or that it's objectively better, I would be suspicious. That's because if we created the mirror image of the Taj Mahal, the right side would be on the left, Jen would not be able to tell the difference and so would say that the left side is nicer than the right one, but it would secretly be the right side she was speaking about. Muwahahahaa, it can't be right. At least if all that determines niceness isn't about where something is placed but what it's composed of - the left side can be nicer than the right, but not just because it's on the left.

Now for three similar examples, each trivial on its own but illustrating a broader point.

1) Most Israeli Jews think Israel is broadly in the right in the Israeli-Palsetinian conflict. Most Palestinians think that the Palestinians are broadly in the right. What bothers me about this is that both factions think that their faction is objectively in the right, that is, independently of who considers the question. But if it's independently of who considers the question, how come people who were born on one side of the aisle mostly think one thing, and people on the other side think another thing. This screams inconsistency and irrationality, since a factor that shouldn't affect what's objectively right affects what most people think.

2) I'm usually more left-wing in my opinions. So when I consider whether some extreme right-wing statement falls within the limits of free speech, I try to think whether I would consider an equally-extreme left-wing statement to fall within the limits of free speech. I imagine a world where that opinion is banned from being expressed, and whether I'd be happy with a world like that. If for the right-wing statement my feeling is "This statement should be banned!" but for the left-wing statement I feel "It would be a terrible world if that statement were banned" then I suspect myself of being biased, since a factor that shouldn't affect what falls within free speech (the alignment of the statement) correlates with my judgment.

3) There's a cliché that everyone driving faster than you is crazy but everyone driving slower than you is an idiot. Since being crazy and being an idiot are traits that are independent of who makes the judgment and many people make different judgments, you should suspect bias, since a factor that shouldn't affect whether someone is objectively an idiot (how fast you are driving) correlates with your belief.

(I'm not sure that craziness and idiocy are really objective. Rather, from inside they feel like they are, and can be defined objectively to accord to that - e.g. if the risk of that person dying outweighs their gain from arriving faster, they are "crazy" drivers.)

These are all examples of cases where I use symmetry to flag beliefs suspected of being biased. This is symmetry in the mathematical sense, of changing some factor which the result should not depend on (nationality determined by birth, political alignment of statement, speed you prefer driving) changes the belief about the result (objective moral judgment, free speech application, insanity/idiocy), and so that belief is suspicious. The important property here is that the result should be invariant to changing something, it's symmetrical with respect to that factor, the same way that the Taj Mahal (and niceness) is symmetric with respect to reflecting it about the middle which replaces the right and left sides.

Before I go on to how I use it, I want to impress upon you that this line of argumentation is very common. Here are examples from SlateStarCodex which is very fond of it, though maybe implicitly. "The control group is out of control" uses the fact that we should judge scientific studies based on their methodology and not the subject matter - our scientific judgment should be symmetric with respect to changing the specific matter studied - and so if parapsychology has the same methodologically-robust results as other fields then if we believe results in other fields we should believe results in parapsychology (and then goes on to argue that since these results are probably false we should be suspicious of the other equally robust scientific results, to preserve the symmetry). "Guided by the beauty of our weapons" advocates for persuasion methods that would lead us towards the truth regardless of whether we are right or wrong when we first use them - weapons whose effects are symmetric with respect to the view of the wielder (which are confusingly called asymmetric in the post). "Against Murderism" shows many arguments that are symmetric with respect to substituting racism for "murderism", and argues that since with murderism we see these are absurd, we should reject them for racism as well.

Lastly, from "Proving too much":

Proving Too Much is when you challenge an argument because, in addition to proving its intended conclusion, it also proves obviously false conclusions. For example, if someone says “You can’t be an atheist, because it’s impossible to disprove the existence of God”, you can answer “That argument proves too much. If we accept it, we must also accept that you can’t disbelieve in Bigfoot, since it’s impossible to disprove his existence as well.”

The argument might work for God, but then it would also work for Bigfoot as well - it's symmetric with respect to replacing God with Bigfoot, since it doesn't use any property which distinguishes them - and so if it proves something about God, it also proves the same thing for Bigfoot.

How I use it

Finding these irrelevant factors that affect our beliefs really helps in raising suspicion of them and not letting them fly under the radar - once you consider, many of these are easily exposed - but we often don't consider. I try to always be aware of irrelevant factors ("symmetries") that shouldn't affect the truth but that might affect my judgment on some matter, and then I try to simulate to see if I would have judged differently if those factors were different, and if so, flag that belief for inspection. For example, whenever I try to judge if free speech applies to some opinion, I flip that opinion about the center and see if it would apply to an equally-extreme opinion on the other side. When I try to form a judgment about tax policy, I try to think what I would have thought about it if I was poorer. Examples of these factors are all the biases we all know and love - sunk costs, cognitive dissonance, the affect heuristic, status-quo bias etc., but also factors that need not have a standard name but apply to a certain situation. There is no "the speed you prefer driving affects your judgment bias", though I'm sure it can be an instance of some bias. 

Finding these symmetries is not easy (if it was, all these SSC posts linked above would have been self-evident, which they aren't), but it's not always very hard if you explicitly try. It's especially hard to find ones you can get good mileage out of, ones that really often affect your judgment but really are symmetries for the outcome. For example, translating the time of judgment by one day and asking "would I still think the same thing tomorrow?" is a symmetry for many judgments but might not help you change your mind since it might not affect your judgment very often - you're as likely to make the same mistake about crazy drivers tomorrow (though it might help with present bias).

And if you find good symmetries for a specific issue you can use them whenever that issue comes up, so they're reusable. I use my symmetry about free speech sometimes, and the one that asks what I would have thought of a law as a person with the opposite interests very often. What are yours?