Depending on how close together the practices are you could maybe try timing your dentist appointment to fit in with your next ortho adjustment, so you go to the ortho & have the archwire removed, then go to the dentist for the cleaning, then return to the ortho to get the wire replaced.
Having said that, that kind of arrangement is completety impractical for me! But I had a cleaning at my dentist's the other day and she was able to work round the brackets and wires no problem.
The only thing she was a bit wary of was spraying the front surface of the teeth (they do a kind of salt spray clean here, akin to sandblasting

which I never encountered in England.) Apparently there can be a kind of lacquer on the teeth which the ortho puts on under the brackets to help avoid discolouration, and which the salt spray might damage. Typically, I don't know if I've got this and my ortho was on holiday so we couldn't ask, so she just got round it by keeping the spray off the brackets. The usual plaque removal, polish etc bit was no problem at all though.
Probably the best thing to do would be to check with your dentist whether they're happy to clean with the archwire in place, and with your ortho if they have any objections/special requirements...
P.S. My dentist told me from the outset that I should have a professional cleaning
at least every 6 months (though I do have dodgy gums) so don't put it off any longer!
