What's the algebraic property where you can flip the fractions in an equation?

This is not an algebraic property, because it is not true.

For example, let $R_2=R_1=2$ and $R_e =1$. Then, $$\frac{1}{2} = \frac{1}{1} - \frac{1}{2}$$ but $$2 \neq 1 - 2$$

You can flip if you flip correctly. Flipping both sides of

$$\frac{1}{R_2} = \frac{1}{R_e} - \frac{1}{R_1}$$

gives you

$$ R_2 = \frac{1}{\frac{1}{R_e} - \frac{1}{R_1}}$$

Well, that's not quite right: more pedantically, flipping both sides gives

$$ \frac{1}{\frac{1}{R_2}} = \frac{1}{\frac{1}{R_e} - \frac{1}{R_1}}$$

but we know that the left hand side of this is the same thing as $R_2$. (at least in the current setting, where $R_2$ is known to be nonzero)

Yes, you can “flip”, so long as the two sides are single fractions: from $$ \frac{1}{R_2}=\frac{R_1-R_e}{R_1R_e} $$ you can rightly deduce $$ R_2=\frac{R_1R_e}{R_1-R_e} $$

Note that, in general, $$ \frac{R_1R_e}{R_1-R_e}\ne R_e-R_1 $$ Indeed the equality would imply $$ R_1R_e=-R_e^2+2R_1R_e-R_1^2 $$ or $$ R_1^2-R_1R_e+R_e^2=0 $$ Since your numbers are by hypothesis non zero, this would imply $$ \left(\frac{R_1}{R_e}\right)^2-\frac{R_1}{R_e}+1=0 $$ or, setting $t=R_1/R_e$, $t^2-t+1=0$. This equality is not true for every real $t$. So your conclusion is really wrong.