Icon prefixes (e.g. hass, mdi)

I’ve seen examples of two different “prefixes” used in the “icon” field of a lovelace dashboard: “mdi” and “hass”. For example “mdi:run” produces a nice icon. In that case, “hass:run” seems to work as well. “mdi” seems to be an obvious reference to “material design” icons, and most seem to work just fine for me (referencing https://materialdesignicons.com/).

Are the “mdi” and “hass” prefixes truly equivalent to each other ? Are there other legal/useful prefixes ?

I working from memory, so may be off a little, but I seem to recall that HA takes a few of the MDI icons and packages these with the hass prefix. So not all of the MDI icons are available with the hass prefix. The hass prefix based icons are stored locally, whereas (if I recall correctly), the mdi prefixed icons are not and may take slightly longer to load.