Before starting I should mention I am using Craig Gidney's tool and some screenshots from it, quirk, which is definitely worths to be checked as himself also has left an answer here, now here is my understanding:
Lets start with some initial state on quirk so we can have an illustrative perception on whats going on:
This is just some random qubit state I've made using combination of three gates. What happens if we apply Hadamard to this?
Well this is combination of "rotation of the sphere about the $\hat{y}$ axis by 90 degrees, followed by a rotation about the $\hat{x}$ axis by 180 degrees" but lets tears this apart using available simpler gates and the definition your are interested:
Identical to applying Hadamard gate itself, we applied $Y^½$ then X. It may not be obvious how we reached from our input to this, so have at look at Y rotation animation and X rotation at quirk
. Before continuing having look at these also would be useful I guess:
X rotation Y rotation Z rotation
Now I guess here is the interesting part, if we apply $Y^½$ again we will reach to this:
But what does this mean? This is equal to X rotation of the input state:
With all these preparations I guess this is the point, if you rotate a sphere 180 degree just by an axis, equal rotations on other axises before and after first rotation will cancel each out and all you need to reach to the initial state is to apply the first axis 180 degree rotation again, just like double applying Hadamard.
In a sense it is like doing rotation against an actual mirror, on the global coordination the real rotation is different from the rotation happening on the mirror (in here $Y^½$ gate) but there is only one reality. Trying rotating an actual ball against a mirror can help to the understanding :)
And of course this is after applying two Hadamard consequently which is equal to the our initial state:
Here is the quirk's model I've prepared for this answer and can be used and tweaked for getting better understanding.
At the end, I should mention that Hadamard itself can be written as a one step 3d rotation, something like rotate(pi/2, 0, pi)
but one step 3d rotations are less intuitive I guess and two step one axis rotation are easier to understand that's why you've found that quote and it is perhaps better to describe it in this way.