particle:

• -ing and -ing (e.g. "coming and going") - as …たり…たり, after the ren'youkei forms of multiple verbs
• doing such things as... - used adverbially
• expresses a command - as …たり…たり at sentence-end, after the ren'youkei forms of a repeated verb

auxiliary verb:

• to be ［たり］ - from とあり, after a noun - archaism ➜ たる
• indicates completion or continuation of an action ［たり］ - from 〜てあり, after the ren'youkei form of a verb - archaism