Looking for open and closed backs to complement my mixing

1st off,
you know not to mix with headphones.
you need trustworthy monitors to mix on.
 
you can check mixes with cans, sure, micro edit, fix phase issues, stuff like that.
but you do not mix on headphones.
 
2nd, if you were to mix on headphones,
you'd need to up your budget by about 5x to get anything close to accurate for mixing on.
 
but like i already said, 
don't mix on headphones.
 
there are lots of nice budget cans to check mixes on,
lots of reviews to read up on:
 
https://www.cnet.com/topics/headphones/best-headphones/
 
https://www.innerfidelity.com/reviews
 
https://www.headphone.com/
 
https://www.head-fi.org/
 
i have and use some akg 240 ii studios to check mixes on, 
they work great,.
 
also uses some Sennheiser CX300 II CX 300 II earbuds, to check and i use them to track.....
work great.
 

The Sony MDR-7506s are a solid choice for tracking, and not bad for mixing if absolutely necessary.  I like the way they sound; bright and a bit edgy but still pleasing.  Sound isolation is decent, and they stay comfortable for long sessions.  They're definitely one of my favorite closed-back headphones, and a bargain at about $80.
 
I also have a pair of Sennheiser HD280 Pros.  They go for about $99.  These are very good tracking cans because they have excellent sound isolation.  But I've never been crazy about their sound.  They sound "stuffy" to me.  And they tend to wear tight and get rather uncomfortable after a while.  Some people love them, though.  It's just a matter of taste.

For good mixing, you really want open-back headphones.  I have a pair of Sennheiser HD600s that I adore.  These go for about $285 and are worth every penny.  The open design makes them sound very airy and natural, offering detailed sound throughout the frequency spectrum.  I understand they might be a little out of the budget you mentioned, but they're worth the extra cost, in my opinion.  They're comfortable, lightweight, and accurate.  

While it's true final mixing should usually done on studio monitors, you can get pretty darned close with a set of cans like the HD600s.  

If forced to monitor occasionally on headphones here is an option that ties in with headphones. I would download the demo and also determine if the plugin in covers your model.
Uses a webcam to track your head. Adjusts EQ for your model. Does some other panning stuff. Quite cool actually.
https://www.waves.com/plugins/nx#introducing-nx-virtual-mix-room
 
The thing I find interesting is we mix on monitors for people who mostly listen on headphones or we mix on headphones and try to make it seem like monitors so that people can listen to it on headphones. Are we going in circles here? If I were serious about a mix I would make it sound good on both, but lean slightly more toward headphones for obvious reasons. We are way past the late 70's when everyone had two speakers in the living room hooked to a stereo.

I would never use headphones for mixing.  To listen to the mix, OK but not for mixing.  
I have some of the headphones previously mentioned and not that fond of the Sony's.  I've got some similarly priced Shure's and they are about the same as the Sony's.  My favorite is an AT ATHM50X for "low price" studio headphones.
 
My take is mix as good as you can get it with studio monitors and others will be good too.  But the right studio monitors.  

Sonarworks or Waves NX used with headphones they support can go a long way to resolving most (but not all) of the problems of mixing using headphones. Especially if the monitors are too small to produce deep bass, or the room doesn’t allow for it.

Checking a mix on cheap earbuds is also a good idea because lots of people listen to music that way most, maybe all, of the time. I suggest getting a set of the bog-standard wired Apple ones, they’re the standard issue with all the iOS stuff, and actually aren’t that bad to listen to, unlike the cheap Chinese knock-offs.

Hopefully no one is intentionally derailing the topic . The pros and cons of using headphones in the studio are pretty clear, I think we all know them... I'm just looking for good recommendations on sets that can help the mixing process.
 
There's a fine line trying to strike the balance between mixing for monitors and mixing for headphones because SOOO much of general listening these days is done on headphones and cheap earbuds... I pretty much NEVER do any pleasure-listening on speakers. Heck, the only pair of loudspeakers I own personally that aren't my monitors in my studio are from an old 90's stereo system from my parents.