Economics is a much less beautiful and elegant field than physics. But it does have some theories that I've always found attractive. I'm going to call these the "It doesn't matter" theories. Here are a few:
In my own field of money we have the quantity theory of money (QTM), purchasing power parity (PPP) and the Fisher effect. According to the QTM, an increase in the money supply leads to a proportional increase in the price level. It doesn't matter for any real variables. According to the Fisher effect, an increase in the inflation rate leads to an equal increase in the nominal interest rate; it doesn't matter for real interest rates. And according to PPP, a change in the relative price levels of two countries leads to an offsetting move in the nominal exchange rate; it doesn't matter for the real exchange rate. (And you can reverse cause and effect here.)
Each theory comes in a strong, semi-strong and weak version. I'll just explain for the QTM, but the idea applies to the other two as well:
Strong: Changes in the money supply and price level are proportional. (Sometimes this is done with the money supply and NGDP, implying fixed V.)
Semi-strong: An exogenous change in the money supply will cause a proportionate change in the price level.
Weak: An exogenous change in the money supply will cause a proportionate change in the price level, in the long run. (That's the version I believe in.)
The same three versions apply to PPP and the Fisher effect.
Of course these three categories bring to mind the Efficient Markets Hypothesis (EMH). It says that it doesn't matter whether you pick stocks recommended by experts or fools, companies that are doing well or are horribly mismanaged, your expected (risk-adjusted) return will be exactly the same in all cases. Noah Smith recently said that while the EMH is technically "wrong," one couldn't imagine an alternative universe where it isn't taught as part of a finance course. I'd say exactly the same about the QTM, PPP and the Fisher effect. They are "wrong" in their strong and semi-strong forms, but also an inescapable part of macroeconomics.
The basic idea behind the EMH applies (more loosely) to other areas, such as compensating differentials in rents and wages. It doesn't matter if you begin farming in an area with good soil and easy access to markets or a not so good area, because any advantage is priced in to land rents. Ditto for wages in dangerous jobs. (Although in this case it may matter a little bit, as people have different preferences for risk.)
I'm not an expert on the Coase Theorem, but is says (approximately) that it doesn't matter where you assign legal liability in externality cases, as long as there are no transactions costs.
I know even less about Modigliani-Miller, but it says something about it not mattering whether firm uses debt or equity financing, at least under certain tax regimes.
Tax theory is chock full of "it doesn't matters:"
It doesn't matter whether sales taxes are applied to the consumer or producer. It doesn't matter whether labor taxes are imposed on the company or worker. (Note that if wages and prices are sticky, however, it may matter in the short run.)
It doesn't matter whether you have a uniform 10% import tax, or a uniform 10% export tax. Ditto for uniform import and export subsidies. And if you have both a 10% uniform export subsidy and a uniform 10% import tax, then you have a big fat nothing. It doesn't matter at all. And yet countries with this set of policies are often viewed at "mercantilist."
It doesn't matter in the long run if you have a flat rate 10% wage tax or a flat rate 10% consumption tax (in the unlikely case wages and consumption are measured properly.)
One of my favorite "it doesn't matters" is the fungibility of money. When I was a senior in college I wanted to buy a car, but couldn't afford it. I only had enough money to pay for tuition. So I applied for student aid, and was awarded $700 (the sum total of my aid for 4 years at Wisconsin.) The aid came with the stipulation that it must be spent on tuition and books, not luxuries like cars. So I abided by this regulation. I spend my own $700 (that I had intended to spend on tuition) on a beat up old Pontiac Ventura, and spend the aid money on tuition. Did I cheat? Who can say? Money is fungible.
Of course the real world is full of complexities that mean it does matter. But even in those cases you need to start from the "it doesn't matter" perspective and then add in realism--otherwise you'll get hopeless lost.
Physicists are obsessed with unifying theories, finding underlying symmetries between the various forces of nature. Much of economics is also the search for hidden linkages, seemingly dissimilar things that are actually alike. I haven't read the famous book by Acemoglu and Robinson, but I gather they tried to unify seemingly dissimilar policy regimes like communism and African or Latin American oligarchies under the umbrella of "extractive." Coase tried to show that there were symmetries between the reason firms exist and the reason pollution is considered a problem calling for regulation. Laffer showed that high tax rates on the rich and the poverty programs for the poor both involved high implicit MTRs. (Explaining, for instance, why rich women were less likely to work that poor women during the 1950s, whereas the opposite is true today.)
These ideas are my favorite part of economics. (I can't imagine there are very many "it doesn't matter" theories in sociology or poly sci.) Deeply internalizing them makes you a better economist, with one exception. Never, ever, fall for the argument that "it doesn't matter" if the Fed swaps zero interest cash for zero interest bonds. It matters very much, as is apparent if you look at how the stock market reacts to new policy initiatives in this area.
However, if the Fed swaps zero interest cash for zero interest bonds, and says (or hints) that the money injection is temporary, then it doesn't matter.