It's well known that there do exist area preserving maps between the sphere and the plane.
It's well known there do NOT exist distance preserving maps between the sphere and the plane.
So naturally one can ask if Hausdorff $2$-measure can be preserved and Hausdorff $1$-measure isn't preserved then is there some $1 < \theta < 2$ such that Hausdorff $\theta$-measure can be preserved.
My expectation is that either measure preserving maps exist for all $1 < \theta \le 2$ or no measure preserving maps exist for $1 \le \theta < 2$. Any other behavior would be certainly surprising and or interesting.
Some Ideas of Mine:
A related question might be to ask for maps from the Sphere to itself which preserve a particular Hausdorff Measure. This question might be a bit more tractable.
Suppose you fix the sphere along an axis $A$. And divide the surface of the sphere up into slices each of which can be independent rotated. So the map of the sphere can be viewed as a map along the axis and at each point of the axis an angle is indicated for how much that particular slice rotates.
We can characterize these maps from the sphere to itself by looking at functions from $[-r, r] \rightarrow [\pi, -\pi]$
As long as ALL the slices are rotated the same angle, you get an isometry which preserves $1$-measure. Which we show below as a green function from axis-space to angle space
Now the minute this green curve is no longer constant, the implied map stops preserving length. We know that for any smooth function we put on here the implied map will preserve area. I want to believe that (and I don't have a proof of this) for any continuous function with hausdorff measure $\le 2$ the implied map between spheres will preserve area. It would be beautiful but perhaps unsurprising if we stretched this idea in 2 dimensions to say that any continuous function with hausdorff measure $1 < \theta < \tau$ from axis to angle space preserves the Hausdorff $\tau$ measure on the sphere. But these ideas are far from rigorous and might even be completely wrong.