Currently it's only possible to Alt + Tab forward through the available windows but some systems also recognise Alt + Shift + Tab, allowing one to go backwards through the list as well.

As a"linux distribution", antiX is a curated collection of components and programs.
For many"feature requests", you would need to contact the (upstream) developer of a specific component.

Ironically, eleventeen times out of twenty (IOW, more often than not)
such feature requests are the result of folks not reading the available user docs
so not realizing the requested feature/behavior already is available (e.g., can be configure via userprefs)

I just checked iceWM (I'm typically using fluxbox) and discovered that its preconfigured alt+tab keybind raises a"window switcher popup".

In fluxbox, not only does a preset keybind exist for"previous window"... IIRC fluxbox provides the flexibility to choose (configure) whether the tab ordering/cycling considers the order in which windows were opened, vs were last visited.

Which window manager, er"desktop session" are you using?

I don't know the answer to your question but I'd like to refine my feature request to say that the default configuration for antiX should make Shift reverse direction for b the tab key just like it does in many situations and not just the alt tab thing. If it's already a feature of the software, I think antiX should be preconfigured to use it.

Lubuntu is what I'm using right now and it functions as expected in this regard.

In IceWM and Fluxbox the key binding alt+shift+tab already works as desired (in IceWM within the window-switcher-popup skidoo mentioned – just repeatedly press and let go the tab key while keeping the other two keys, alt and shift, pressed).

For JWM insert the line
into the file ~/.jwm/keys and restart JWM.

EDIT: Just discovered that IceWM lets you switch windows without opening the popup: alt+esc gives you the next window, alt+shift+esc the previous window!