Does what you're hearing in the DAW always what you hear in

Import the track into your DAW and flip the phase. You'll get a 100% objective answer that way.

Did itunes sneak in the spacial audio setting on you?  I feel like with every update I have to go turn that shit off again.

Once upon a time, there were people who swore that you had to bounce down realtime and send that out thru a pair of DA, and then either back into a pair of AD or into a standalone box capture recorder.

I get that, by doing that, you eliminate any possibility that some funny business is going on, because what you're hearing while monitoring real time is also what is getting recorded as a final mix down. Once upon a time, before DAWs, this was obviously the only way you could do it, but I wonder if such an approach is still relevant in DAWs in 2024.

I have a Tascam DA-3000 for standalone mixdown capture, so I can and do use this method anyway, but I have wondered about all of this sort of stuff from time to time.

This is what I do, recording back to a print track inside PT via internal busses with hardware on them.  Always sounds identical.

Print mixes here too.

But there is also probably some psychological aspects of listening while working in the daw vs not.

Cubase.

Render offline
Record back into a track.

Both null to silence with the original.

So speaking for Cubase mixes are identical when mixed down.

What can change is when you Master and start raising levels with more compression and limiting as then the ratios of ambience
etc start to change so I always throw a limiter plugin (just for monitoring purposes) near the end of a mix to see how the mix is going to
survive the next stage of production.

I pretty much do the same, except I route the "roughs" to a dedicated pair of DA that input separately into my Xmon monitor controller.  That way I can switch between the rough and current mix there.

I always print in real time back into the session.  Sounds the same.

I wouldn't trust iTunes as far as I could throw it.  Can you play it back through the Pro Tools browser/workspace?  that always sounds the same for me.

There have definitely been times in the past where plugs have had issues with full speed offline renders.  Some of the highly oversampling ones and reverbs have had issues for me a decade ago, but I haven't run into any that have those issues anymore. 

I can do full-speed offline and 1x online renders and they sound identical these days.  

One thing I still do is only render out to full quality and then do a batch conversion to get the MP3 and CD quality versions from the HD version that I rendered, but I do this mainly because I have batch settings to do it quickly.  The conversions seem to be a lot faster than the offline render overall so even though it's a sidestep process, it's faster than rendering twice more.

Usually no using Reaper and Foobar, but I have experienced what Dan mentioned where a limiter will behave differently (missing a transient that it caught previously or vice versa) on separate, full speed passes.