Most PDF readers (as in, all but the most rubbish ones) have the option to consider the cover as a separate page when viewing two pages side-by-side. In other words, you can choose to have the odd pages on the left or on the right side. There's no need for a version "optimized for two-page on-screen viewing".
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In Acrobat Reader, it's in the menu "View" -> "Page Display" -> "Show Cover Page in Two Page View".
Unfortunately, in Acrobat Reader there's no button and no keyboard shortcut for it, you got to use the menu. That's one reason why I prefer other PDF readers (PDF-XChange Editor on Windows and Foxit on macOS).