What are the most general and powerful math tricks? I quote from the first chapter of Michael Spivak's Differential Geometry Volume I:
The precise definition of $\mathbb{P}^2$ uses the same trick that mathematicians always use when they want two things that are not equal to be equal. The points of $\mathbb{P}^2$ are defined to be the sets $\{p,-p\}$ for $p \in S^2$...
That passage burned itself into my brain when I first read it decades ago.  My immediate reaction was, "I want the list!  Where is the list of tricks like that?!"
In my studies I have kept an eye out for tricks that generalize, that carry over into a different area.  I also watch out for very broad categories of trick that have different specific versions in different topics.  I tutor math and physics and have compiled lists of "debugging" steps for when a student is stuck on a problem.  But I have long suspected that there are far more powerful tricks out there that are common knowledge to mathematicians but not to mathematics students.
In an effort to clarify what I am asking, I will write my own best answer, and I hope others can improve, elaborate, or best of all add completely new items to the list.
 A: The most general math tricks I know include

*

*Add zero. An example of this is completing the square.  Faced with $x^2 + 6x + 1$, we can add $0 = 8-8$ to produce $x^2 + 6x + 9 - 8 = (x+3)^2 -8$.

*Multiply by one. The simplest example of this is addition of fractions.  The most common example is probably unit conversions. Other versions include the "conjugate trick" (see below).

*Factoring an unknown. The solution to the cubic equation involves replacing the single variable $x$ with $u-v$, where you now have two variables, and at a later point can use the extra flexibility to declare some messy quantity to equal zero.  A similar technique is variation of parameters in differential equations, where an unknown function is declared to be a different unknown function multiplied by a known one chosen for convenience.

*Separation of variables. Independence in probability is one case, and another is in partial differential equations, writing the solution as a product of solutions depending on variables separately.  Arguably that is a special case of #3.

*The "Spivak trick" quoted in the question.  I have rarely seen this used but it seems powerful and important.

*The conjugate trick. A special case of #2, this is used extremely often, for rationalizing a denominator in fractions, complex arithmetic, L'Hopital's Rule, and more.

*Symmetry.  I don't even know how to characterize this.  While I make significant use of it, there is often yet more symmetry that can be exploited to cut a problem in half, declare an answer zero, and so forth.  It is clearly extremely powerful. But, for example, I once read a 12-page proof that a quintic equation does not have a general solution formula, which seemed to be 11.5 pages of group theory that seemed to have absolutely nothing to do with the question, followed by some handwaving and declaring "QED".

*Solving a more general problem. This seems to require the most creativity, and so is hard to really classify as a tool more than a hope.

This is the broadest classification I know.  I have constructed this mental model of techniques slowly over years, and I am curious what other powerful yet comprehensible techniques exist.
