Assuming I am working on 3 screens, having 3 spaces on each screen.
How can I switch to the "left" space on each screen at the same time?
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Only by not using System Prefs > Mission Control > Displays have separate Spaces.
Once you do that, all 3 screens will 'share' the same space & switch together.
Personally, I find it a much better way to work, but I guess it will depend on personal preference. Using this method I like to 'tie' certain sets of applications together so I can see each 'set' on a different Space.
Each time you swap methods, all your carefully-prepared Spaces will mess up, as the numbering will change, so be prepared to have to reset which apps are set to open on which Space, if you wish to try it out.