[santra]

OK, I think I sort of get it.

The "Due" Perspective on the desktop shows:

Due within the last 3 months
Due within the last week
Due tomorrow
No due date

The "Due Soon" Perspective on the iPhone just shows upcoming due tasks (according to a number of days specified in Settings on the iPhone).

I used the "snapshot" icon on the Perspectives menu to tweak both the "Due" and the "Overdue" Perspectives to my satisfaction.

Success!