In my experience, both good headphones and monitors are a great combo for both the mixing and mastering processes. The reason I said "good headphones" is because I find anything less than a good headphone is counterproductive, whereas monitors don't need to be stellar to get great results.
In the case of monitors, what's important is to have a fairly revealing monitor so that mix problems are not masked too heavily by the inadequacies or compromises in their design. It's OK that monitors themselves are imperfect, because so long as we can clearly understand their imperfections, we can mix accordingly. Take the HHB Circle 5 as example; a mid-range, mid-price passive midfield from some years ago. A number of producers I know swear by them, even in the face of much nicer alternatives. I've auditioned countless mixes and a great deal of music on them. I know for example that they tend to roll off in the bottom-end, and have a big dip in the upper mids. So I can mix expecting that sound, and know that those mixes will translate well to other equipment. In my book, regardless of the their imperfections, this makes them a "good" monitor speaker: predictable, understandable and easy to create good mixes on; so long as your ears and brain can understand their characteristics, they're not hiding too much away from you to produce a good mix. I can hang the same label on eg. monitors from Adam, Mackie and some of the K-Rok's. None of these monitors are big money, but all allow for the possibility to create great results if your ears and brain can "learn" them. I can't say the same for (at the cheap end) anything Behringer or Tannoy, or (at the more expensive end) anything JBL, though I'm not familiar with their latest models. We've all heard stories about engineers from back-in-the-day swearing by NS-10's, well guess what: they're not a great loudspeaker but they're still a usable monitor, particularly for material that doesn't have the widest dynamic range (140hz thru 14k they're rolling off heavily at either end). The rule with monitors is, if you can get to know and understand them, they're a good monitor for you.
With headphones though it's quite a different story. Nearly all headphones colour sound and mess with it heavily. Most budget cans are very forward in the mids and up, don't handle transients at all accurately and do all kinds of trickeries to "enhance" the bottom end, none of which is great for mixing. That doesn't mean they're bad cans, they're just bad
for this job. For example, the classic Beyer DT-100 is a great budget headphone for vocal foldback, but awful for mixing. There are always exceptions though, and probably the only cans I've found that are really
good for this are Sennheiser HD580/600/650/800. None of these are cheap, though you might find a few bargains on the 580. The design of these headphones is transparent sound, very little messing around with the source, tight bottom end, very smooth all the way up to 30k (that of it which I hear anyway!) and exceptionally fast and revealing transient response. This makes them great for certain kinds of work when mixing and mastering. For example, adjusting compressor attack etc. on basses and drums. I can get this right on the cans instantly, but on the monitors I'm always fiddling. Headphones tend to goof around with the soundstage, so for positioning instruments in the Z-plane (setting verb levels and such), they're a pain, ditto for panning placement. But for identifying little problems in the mix that sneak by on the monitors, they're great; little clicks, or noise in-between take items etc..
In sum, the combo of both is very capable with the right headphones and reasonable monitors. I liken the HD600 to how I would use small-box nearfields; they do the same job for me. You can address all the issues in your mix and master, hopping between cans and monitors until you've nailed everything. It's probably not the fastest way of working, but I seem to get great results with this method and it's become my go-to workflow.
Edit: I would add that if I had to choose only one it would be monitors every time. I would never mix/master entirely on headphones, regardless of the model, since there's too much that can go wrong. I tried it a few times and had some shock moments when I later listened to my work on monitors and various hifi setups!
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