One of the most irascible words in the English language is the American neologism, irregardless, popularized during the 20th Century and meaning: without consideration; or, not needing to allow for; or, heedless; or, without reguard. The word is a combination of both irrespective and regardless, which raises a rather peculiar problem, namely, that both of the base-words are synonyms, thus, engendering a double-negative.
Since both irrespective and regardless mean, roughly, “heedless,” when one is saying irregardless, what one is actually saying is heedless of heedless, or, more exactly, irrespective of disregard OR without without reguard.
Since here, one is disreguarding their disreguard, they are, in effect, maintaining their reguard (provided they had some to begin with), however, this is the precise opposite of the meaning entailed in the casual (as opposed to literal) usage of “irreguardless.”
The solution to this arcane conundrum is, thankfully, quite a simple one: don’t use the word. Both irrespective and regardless are synonyms to this confusing adverb, and hence, can take its place without any linguistic confusion UNLESS one is writing fiction and one is emulating a certain regional dialect.