AFAIK SQL table indices are to improve queries speed, so are not absolutely necessary. However SQL Azure for whatever reason demands that each table has a clustered index, so I guess there're cases when an index is necessary.
What can be a case when a clustered index is absolutely required and why?
The absence of a clustered index implies a completely different storage model for the table, namely a heap table. I suspect that the SQL Azure team simply didn't implement this, and therefore tables must be stored in the "regular" way.
This article describes some of the pros and cons of heap tables and clustered tables.