This odd preposition can also appear as a postposition, but does not behave much like either.

Something of a syntactic nut, notwithstanding can appear as either a preposition or postposition:
notwithstanding your preferences in the matter, we will foreclose
your preferences in the matter notwithstanding, we will foreclose.

The OED tells us that it consists of not and withstanding, and follows the mediaeval Latin pattern of non obstante to operate as

• a preposition, meaning in spite of (from around 1380),
• a postposition, following this, that or a noun, differing only in order from the previous (1490)
• an adverb, meaning nevertheless, still, or yet (1440),
• a conjunction, meaning although, where it can be followed by that with a dependent clause (1449).

(I will not concern you further with any adverbial or conjunctive uses.)

In turn, withstand is a verb derived from Old English, meaning to stand resistant against (as a transitive), or to resist (intransitive). So negated its original sense would be ‘not resisting’, which is a rum start for a preposition.

Delve a little deeper into patterns of its use as a pre(post)position, and it gets stranger still.

Diversion to Hamelin

Bear with me for a little diversion into stranding and pied-piping (linguists who understand these already can skip to the final section).

Normally a preposition remains directly in front of its object (the entity to which it is attached), as in notwithstanding your preferences.

However in certain constructions, such as in questions and with relative clauses, word order becomes altered, and the preposition becomes separated from its object,
you talk about words becomes which words are you talking about?
which strands about at the end of the sentence. Although perfectly natural, many grammar peevers have strange fetishes about prepositions being stranded at the end of a phrase.

The alternative is known as pied-piping (after the Pied Piper of Hamelin), in which the whole construction is shifted around. For example,
Adie spoke with Belinda – standard order, declarative
Who did Adie speak with? – stranding, question
With whom did Adie speak? – pied-piping, question.

Back to notwithstanding

John Culicover has pointed out that notwithstanding can only undergo pied-piping when a preposition, and does not strand too well either. From:
we are going to demolish the building, notwithstanding your very generous offer (preposition)
we are going to demolish the building, your very generous offer notwithstanding (postposition)
we might re-order to:
that was a very generous offer, notwithstanding which we are going to demolish the building (preposition, pied-piping)
*that was a very generous offer, which notwithstanding we are going to demolish the building (failed pied-piping)
?that was a very generous offer, which we are going to demolish the building notwithstanding (questionable stranding).

It is hard to come up with any other word order which is as comfortable as the pied-piped preposition.

Culicover considers whether it might be better to pretend that notwithstanding was some kind of adjective, but that does not hold good. The suggestion that it is actually a participle, as it might given its etymology, is interesting, but participles strand, do not pied-pipe, and cannot be postpositional.

So the only conclusion is that notwithstanding is an unusual preposition-like element which can act as a postposition, but does not strand.

One final twist to this is that a very similar grammatical element, which has similar meaning, is disregarding. However that cannot appear postpositionally, allows pied-piping, but does not strand. Thus notwithstanding looks like another addition to our growing bag of syntactic nuts.

Reference: John Culicover (1999) Syntactic Nuts, Oxford UP, pp 69-71.