Difference between clustered index and partitioning column
But does the partitioning column have anything to do with order on the disk? 


From Clustered Index Structures :


  Clustered indexes have one row in sys.partitions, with index_id = 1 for each partition used by the index. By default, a clustered index has a single partition. When a clustered index has multiple partitions, each partition has a B-tree structure that contains the data for that specific partition. For example, if a clustered index has four partitions, there are four B-tree structures; one in each partition.


From Table and Index Organization:

When a table or index uses multiple partitions, the data is partitioned horizontally so that groups of rows are mapped into individual partitions, based on a specified column. The partitions can be put on one or more filegroups in the database. The table or index is treated as a single logical entity when queries or updates are performed on the data.

The pages in the data chain and the rows in them are ordered on the value of the clustered index key. All inserts are made at the point where the key value in the inserted row fits in the ordering sequence among existing rows. The page collections for the B-tree are anchored by page pointers in the sys.system_internals_allocation_units system view.

If you want to really dig more into how data is layed out then refer to : Inside The Storage Engine: sp_AllocationMetadata

As a side note : SQL Server 2016 CTP2 has TRUNCATE TABLE ... [ WITH ( PARTITIONS ( {  |  } 

For automating partitioning switching check out - SQL Server Partition Management utility