Does it matter if I reverse the arguments order of a lambda calculus function?
You can compute either of them from the other, by using the aforementioned "reverse arguments" function.

However, they are very different as functions. For instance, the usual encoding of true is $\lambda xy.\, x$ while false is encoded as $\lambda yx.\, x$ ($\alpha$-equivalent to $\lambda xy.\, y$), which is the flipped version of true. True and false are meant to be different.

Indeed, if we had true and false equal then, for any $M,N$
$$
M =_\beta^2 (\lambda xy.\, x) M N =_{Hp} (\lambda yx.\, x) M N =_\beta^2 N
$$
So, we would get that all $\lambda$-terms are equal, making the whole theory of lambda calculus trivial.