"Sir or Madam" vs "Madam or Sir" in formal letter

In a formal letter addressed to one or more unknown recipients, "Dear Sir or Madam" is the customary salutation. As a German native speaker, who is used to "Sehr geehrte Damen und Herren", writing "Sir" before "Madam" sounds impolite to my ear and I feel the urge to change the order. I know that I must never directly carry over conventional expressions from one language to another, but it makes me wonder:

Are there any differences between "Dear Sir or Madam" and "Dear Madam or Sir"? In particular, would the latter sound more polite or rather come across as outlandish?

In English, "Dear Sir or Madam" is the traditional and customary order. It does sound quaint (and sexist to some) — but's that's how it is.

You would indeed be thought "outlandish" (a good choice of term there) if you were to reverse that order.

While I realize the traditional way to write it is with "Sir" first, but I practice writing most anything requiring a choice in order alphabetically. Thus, I write "her/him," "he/she" and even lists such as: "charismatic, determined, enthusiastic, influential and multi-faceted" in alphabetical order.

Sometimes, the words I'm alphabetizing seem out of order and yet, as with anything new, I expect that to be the case. Thus, I endure the discomfort and write them as I've reasoned is optimal until such feels more comfortable, which it inevitably does, save some overt flaw in my reasoning. Oh, to an earlier reference about "ladies and gentlemen," I'd opt for "gentlemen and ladies," for reasons herein explained.

I believe that this type of practice constitutes progress (even if minute) in our overall work toward gender equality and so it pleases me. I realize that some may find it confusing or off-putting, which only means that they are thinking about it and if enough of that occurs, then we all just might get to a better place.

Dear Sir or Madam is the customary ordering, which is reason enough to do it in a formal setting. But also consider the "rhythm" of the opening; sir has half the syllables as madam, and therefore it sounds better when sir is uttered first. Same thing with Ladies and Gentlemen, where ladies has less syllables than gentlemen. Compare this to German, where Damen and Herren have an equal amount of syllables. Now, I (as an ESL writer) am not sure if this effect actually carries over to writing, though it is something to consider.