Subtraction is harder than addition, division harder than multiplication, integration harder than derivation and, in general, finding a needle is harder than hiding it. In his annotations for Newton's Method of Fluxions, John Colson (the fifth Lucasian) wrote that in the Science of Computation all the Operations are of two kinds, either Compositive or Resolutative. The Compositive or Synthetic Operations proceed necessarily and directly... and not tentatively or by way of trial. Such are Addition, Multiplication, Raising of Powers, and taking of Fluxions. But the Resolutative or Analytical Operations, as Subtraction, Division, Extraction of Roots, and finding of Fluents, are forced to proceed indirectly and tentatively, by long deductions... and require the contrary Synthetic Operations to prove and confirm every step of the Process.
This is philosophically very relevant, since, despite being a synthetic operation and its analytic partner mutually inverse, their difficulties are painfully asymmetric. Like going up or down a slope. The processes look symmetric (doing one and then the other results in nothing), yet the difficulty of going uphill, at least from a cardiovascular point of view, is greater. This is, in part, because symmetry is broken by gravity, pointing down and not up. So, since analytical operations are much harder than their synthetic counterparts, is there some 'gravity' in them as well?
We can be even more picky and say that the gravitational potential is conservative, so that the process of going first downhill and later uphill should result in zero work done. But alas, our muscles operate under heavy friction, and heat will be produced in the process, i.e. energy will dissipate, irreversibly. So symmetry is also broken by the thermodynamic arrow of time.
What is the nature of the asymmetry between synthetic and analytical operations in mathematics? I have no clear answer for that. Maybe there is one and I just don't know it. Synthetic operations are somehow local, while analytical processes have global properties. To hide a needle in a haystack you just move a step, then another and another, until you place the needle somewhere. If you are asked to find the needle instead, you need non-local information (the history of your steps). Of course, you could locally brute-force an entire sweep of the haystack, but such approach rapidly becomes impractical.
With mathematical operations, synthetic processes consist in having known ingredients, and then producing a result out of them. With analytical operations, you are given the result and you need to guess the original ingredients. One difficulty is that there may be more than one set of ingredients that give the same result, so some restrictions are usually given (like boundary conditions). Another is that there may not exist such a set! But the main one is that somehow you need to consider all the possible ingredients in the world and their possible interactions. The most popular example is the multiplication of two prime numbers (synthetic) vs the factorisation of the product (analytical). The asymmetry is so strong that it forms the basis of public-key cryptography. In other words, in a synthesis you don't lack information, whereas in an analysis you do.
Locally, doing and undoing are reversible. Globally, doing is easy and undoing is hard. Both processes are apparently symmetrical, but they are not, because at every visible step, there is a hidden step that acts as a toll. Doing is going downstream along the cosmic river of time. Undoing, if we could reverse time, would be downstream as well. But, alas, we can't reverse time, and that is why, when you try an analytical operation, you need to swim upstream, against the cosmic flow. When doing a division, for example 17/9, you begin by exploring different downstream possibilities, i.e. multiplications: 1·9=9, 2·9=18, and in this case you stop here (and if clever, you choose 2·9=18) and keep operating with a (hopefully negative) remainder. So, to swim upstream you need to explore a set of downstream paths and choose one.
I don't know of any formal theory that quantifies the entropy production of synthetic operations, but clearly, the result of a sum or a multiplication has more entropy (or perhaps more free energy, a broader concept) than its ingredients. Exactly like in a chemical reaction. At school, inverse operations were presented to me with an emphasis on their dual character, but never with a discussion of their asymmetry. I was taught that these games were in balance, like Newton's action-reaction law, or like the yin and yang. Only to (un!)learn later that such balance was deeply skewed. The yin-yang symbol is symmetrically depicted, which heavily misguides the reality of a yin gigantically greater than the yang (which should be denser, to compensate).
The concept of duality, thanks to time, is fundamentally asymmetric: disorder (chaos) is cosmically bigger than order (cosmos). Lies and the unknown dominate over truth and knowledge. And darkness is basically the queen of a universe where light is spatially anecdotal (and anecdotally brilliant). In Star Wars, the Sith are completely unnecessary and misleading (although narratively understandable): the light side, in order to balance the universe, needs to be highly concentrated in some places, hence the few Jedi here and there. But the dark side is not a side, it is the entire canvas, and as such is diluted. In other words, the same events could have happened by only considering the accumulated stupidity of the anonymous majority.
Summarising: inversion, balance or duality do not imply symmetry in space, shape, size or difficulty. On the contrary: this universe has time in it, and asymmetry is its hallmark. Mathematics, detached from the natural world and supposedly free from the subjugation of time and supposedly being more fundamental than its younger sibling, statistics, still features a landscape with its own irreversible dynamics, i.e. with time and statistics emmbeded on it. This literally blows my mind.