Chirality presupposes some sort of motion or transformation (of the objects together with the surrounding space) to superimpose the left and right handed objects upon each other. It can be either a mirror-reflection that in one step exchanges every point with its chiral counterpart, or a smooth physical motion (requiring one extra dimension, such as placing the 2-d surface of one hand against the 2-d surface of the other by moving it through three dimensional space).
Inside and outside are perfectly sensible concepts mathematically but it is not that common (in particular there is nothing as generic as "inside" and "outside" themselves that would tend to apply when those words do) to have a transformation that exchanges the two. That is true even if one is allowed to distort distances by stretching and twisting the space unevenly. There are operations like inversion in elementary geometry that, loosely speaking, exchange inside and outside of a circle, but these really are switching the in/outside of a punctured circle, or an annulus, which in this context is an after-the-fact complication of the original geometric figure to make the transformation "work" (*), and not a manifestation of any fundamental interchangeability of the two sides.
The transformational picture makes some sense for local orientation. A curve in the plane, a surface in space, and generally an n-dimensional surface in (n+1)-dimensional space, will locally (that is, near any point of the surface, or in small regions) divide the space into a "left" and a "right" side relative to the surface. If there is a global division of the space into parts inside/outside the surface, this will coincide with a notion of local left/right orientation for the surface (one direction will point IN and the other OUT, near any point). If you place this picture into a space one dimension higher, such as drawing a circle in the plane and placing the plane into a 3-dimensional surrounding space, then the local arrows that point "left" and "right" in the plane at each point of the circle, can be rotated in the additional dimension until they are exchanged. There is another type of transformation between left and right in the case of non-orientable surfaces like the Moebius strip. There, you can switch left and right by following particular types of loops through any given point, like an ant walking the length of the Moebius strip, and finding that when it returns the notion of left and right (or up and down, from ant's point of view) have been reversed.
One difference between these transformations and the case of chiral hands, gloves and socks in 3-dimensional space is that for the familiar physical objects, the space and the objects are directly moved by the transformation. In the case of local orientations and higher dimensional placements of the object, the placements and orientations are a sort of extra structure added to the object at each point, and the transformations are moving the structure around while keeping the object in place.
There was a much upvoted earlier discussion here, of topological inside-out transformations and how to formalize them: Why can you turn clothing right-side-out?
(*) by making the transformation "work" I mean that the inside and outside generally are topologically different, so no transformation of the space can make them truly equivalent without some modification of the figure. For example, the outside of a circle or a sphere looks (topologically, i.e. with stretching allowed) like the inside with a solid piece removed. After removing a similar piece from the inside component it becomes topologically the same as the outside, creating the potential to use an inversion or some sort of continuous flow of the space through one higher dimension, to interexchange the two regions. Such a transformation cannot be a distance-preserving operation like a mirror reflection, so even if you allow this sort of modification of the original figure, there is no perfect analogy with the intuitive motivation of an orientation-reversing, but distance-preserving, correspondence between left and right handed gloves, socks, shoes or sugar molecules.