In Maya, variants are represented in the Outliner with a small v icon next to the associated prim. You can right-click that prim and select Variant Sets to change the active variant being composed. For a more robust view of variants, you can use Windows > USD Tools > USD Variant Manager (or by right-clicking a prim with variants in the Outliner and selecting USD Variant Manager).

This view allows you to see all your variants at once in a nested interface. You can switch the active variant by selecting it in the USD Variant Manager, or right-click (or use ⋮) to add or remove variants.
By default, the Variant Manager will update whenever you change selection in the scene (such as when selecting different components in the Outliner). You can use the Pin button lock it to its current state.
You can convert selected objects in the Viewport into a single USD variant set at the time of stage creation. You can also add new variant sets on a stage that already exists.
To create a new USD stage with selected objects as variants:
Maya creates a new USD stage with each of the selected objects as members of a single variant set under the root prim.
Maya will still create a default variant set even when you create a stage from a single selected object using USD > Create Component.
To create a new variant set on an existing stage:
If a component is open as the root of the stage (in other words, not referenced as part of a larger composition), you can use the USD Variant Manager to add / remove variants to it.
To add a new empty variant to a variant set:
Maya adds an empty variant to the variant set.
To add Maya objects to an existing variant:
You must add Maya objects to an existing variant.
To remove a variant / variant set:
Maya removes the variant / variant set and all its contents from the stage.
If a component is open as the root of the stage (in other words, not referenced as part of a larger composition), you can target specific variants in the USD Variant Manager to receive edits (such as changes to geometry, materials, and so on).
To target a variant:
Any subsequent changes you make to the USD component will now be applied to that variant until you change the target.
If you apply an override in a nested variant and don't see the changes composed in the scene, check the USD Layer Editor to see if there is a stronger local opinion overriding it.
A Purpose can be assigned to existing USD prims directly within the Variant Editor using the "Add Selected as Purpose" context menu.
How it works
The selected USD data is automatically moved under the appropriate purpose scope. If the scope doesn't already exist, it will be created for you.
This workflow applies to both Maya data and USD data, giving you a consistent way to organize variants by purpose without manually reparenting or moving prims.
The context menu option is disabled when a purpose scope itself is selected, since scopes cannot be nested inside other scopes.