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Some of the things listed in the Due perspective should have a ">" at the right end of the line, and tapping on those lines should give you another view that shows the actions as if in project mode.

I created a simple project ("Test due project") with a number of due actions (and the project itself also has a due date) as an illustration.

Here you see the project itself showing up after all of its contained actions in my Due perspective:

And here you see what clicking on the project from the Due perspective gives, which is identical to what you would see if viewing it from the Projects view, except the back button is different, of course: