You can accomplish this using Pivot Variables + Calculated Members.
You will need to define a pivot variable named "CommonDate" (or whatever, we will use "CommonDate" for this example). This Pivot Variable will contain values like "65211". The Patients Cube in SAMPLES contains an example YEAR Pivot Variable.
You will then define two calculated members:
1) Start Date = "[StartDate].[H1].[StartDay].&[$variable.CommonDate]"
2) End Date = "[EndDate].[H1].[EndDay].&[$variable.CommonDate]"
You can then create an "Apply Pivot Variable" control in your dashboard pointing to the "CommonDate" pivot variable. This date will then be applied to both dimensions even though they have different names.