functional equations - Strange Examples of Involutory Functions?

I am very interested in functions $\gamma:\mathbb R\to\mathbb R$ with the following property:
$$\gamma^2(x)=x$$
One form of a function satisfying this is
$$f(x)=\frac{a-x}{1+bx}$$
Which has the property $f^2(x)=x$. Infinitely many more functions with this property can be obtained by finding some other injective function $g$, its inverse $g^{-1}$, and then composing $g, g^{-1},$ and $f$ as follows:
$$g^{-1}\circ f\circ g$$

However, I am not very interested in involutory functions of this form, since they seem to all be ripoffs of the general form that I already stated.

In fact, it seems that all involutory functions can be put in the form
$$g^{-1}\circ f\circ g$$
for some $g$, and for some $a,b$. I can't find any counterexamples, but I don't know how to prove it either. It seems to me that the best way to approach this would be to set up some kind of differential equation like
$$(f'\circ f)(x)=\frac{1}{f'(x)}$$
But I have absolutely no idea how I might show that any involutory function can be put in the aforementioned form.

Any ideas?

NOTE: I'm sure there are some elaborate piecewise-defined answers that can destroy my conjecture. However, I can't expect people to know what I mean when I ask to prove this for all "reasonable" functions - so I will establish some stricter restrictions on $\gamma$. The function must be expressible using some finite composition of these functions and their inverses:
$$\phi_1(x,a)=x+a$$
$$\phi_2(x,a)=ax$$
$$\phi_3(x,a)=x^a$$
$$\phi_4(x,a)=a^x$$
For example, $x^2+x+1$ can be expressed as
$$\phi_1(\phi_3(x,2),\phi_1(x,1))$$