stats_column_id and index_column_id do not update with physical order of clustered index is changed Unless I'm misunderstanding the purpose of the column, the following code indicates that a change of the structure of the clustered index does not change the ordinal position (stats_column_id)of the column in the sys.stats_columns DMV. (Tested in AdventureWorks2014, AdventureWorks2008R2)

select i.name, c.name, ic.column_id, ic.index_column_id
from sys.indexes i 
join sys.index_columns ic
    on i.object_id = ic.object_id
    and i.index_id = ic.index_id
join sys.columns c 
    on i.object_id = c.object_id
    and ic.column_id = c.column_id
where i.name = 'PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID'
order by ic.key_ordinal;

select sh.name,s.name, c.name, c.column_id, sc.column_id, sc.stats_column_id
from sys.stats s 
join sys.stats_columns sc
    on s.object_id = sc.object_id
    and s.stats_id = sc.stats_id
join sys.columns c 
    on s.object_id = c.object_id
    and sc.column_id = c.column_id
join sys.tables t 
    on s.object_id = t.object_id
join sys.schemas sh
    on t.schema_id = sh.schema_id
where s.name = 'PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID'
order by sc.stats_column_id;

dbcc show_statistics('[Person].[BusinessEntityAddress]','PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID') with density_vector;

ALTER TABLE [Person].[BusinessEntityAddress] DROP CONSTRAINT [PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID]
GO

ALTER TABLE [Person].[BusinessEntityAddress] ADD  CONSTRAINT [PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID] PRIMARY KEY CLUSTERED 
(
    AddressID ASC,
    [BusinessEntityID] ASC, 
    [AddressTypeID] ASC
)
GO


select i.name, c.name, ic.column_id, ic.index_column_id
from sys.indexes i 
join sys.index_columns ic
    on i.object_id = ic.object_id
    and i.index_id = ic.index_id
join sys.columns c 
    on i.object_id = c.object_id
    and ic.column_id = c.column_id
where i.name = 'PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID'
order by ic.key_ordinal;

select sh.name,s.name, c.name, c.column_id, sc.column_id, sc.stats_column_id
from sys.stats s 
join sys.stats_columns sc
    on s.object_id = sc.object_id
    and s.stats_id = sc.stats_id
join sys.columns c 
    on s.object_id = c.object_id
    and sc.column_id = c.column_id
join sys.tables t 
    on s.object_id = t.object_id
join sys.schemas sh
    on t.schema_id = sh.schema_id
where s.name = 'PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID'
order by sc.stats_column_id;

dbcc show_statistics('[Person].[BusinessEntityAddress]','PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID') with density_vector;


However, the density vectors indicate a change in the leading column of the index/statistics object. Is this a fundamental misunderstanding on my part? If so, how would I find the leading column of a statistics object using DMVs?

Tested SQL Server versions: 2008R2, 2014
I was having the same issue while trying to reproduce the way others retrieve index information from the sys.dm views in SQL Server. I just couldn't figure out the order of the columns in the index. 

Following is a script I created to determine the order of the columns in any given index for a given table:

SELECT s.name                  AS Schema_name,
       o.name                  AS Table_Name,
       i.type_desc             AS Index_Type,
       i.name                  AS Index_Name,
       c.name           AS Table_Column,
       i.fill_factor           AS Indx_Fill_Factor,
       ic.key_ordinal          AS [Key_ordinal (IDX Column_Order)],
       ic.index_column_id      AS Index_column_id,
       stc.stats_column_id     AS Stats_Col_ID,
       -- Additional info for each joined table
       -- comment out what you don't need
       -- 2 lines at a time
       --
       -- '| table object --&gt;', -- column seperator
       -- o.*,
       -- '| schema object--&gt;', -- column seperator
       -- s.*,
       '| index info--&gt;', -- column seperator
       i.*,
       '| sys index info --&gt;', -- column seperator
       si.*,
       '| indx cols info --&gt;', -- column seperator
       ic.*,
       '| tab cols info --&gt;', -- column seperator
       c.*,
       '| idx stats info --&gt;', -- column seperator
       st.*,
       '| idx stats columns info --&gt;', -- column seperator
       stc.*
FROM   sys.objects             AS o
       JOIN sys.schemas        AS s
            ON  s.schema_id = o.schema_id
       JOIN sys.indexes        AS i
            ON  i.object_id = o.object_id
       JOIN sys.sysindexes as si
            ON  si.[id] = i.object_id
            AND si.indid = i.index_id
       JOIN sys.index_columns  AS ic
            ON  ic.object_id = i.object_id
            AND ic.index_id = i.index_id
       JOIN sys.columns        AS c
            ON  c.object_id = ic.object_id
            AND c.column_id = ic.column_id
       JOIN sys.stats          AS st
            ON  st.object_id = i.object_id
            and st.stats_id = i.index_id 
      JOIN sys.stats_columns  AS stc
      ON c.column_id = stc.column_id
      AND stc.stats_id = st.stats_id
      AND stc.[object_id] = o.[object_id]
WHERE  1=1 
     --and i.type &lt;&gt; 1 -- Exclude Clustered Indexes. 0 = Heap; 1 = Clustered Index, 2 = Non-Clustered Index
       AND s.name != 'sys' -- Exclude sys items
       and o.name = 'BusinessEntityAddress'
       AND i.name = 'PK_BusinessEntityAddress_BusinessEntityID_AddressID_AddressTypeID'
ORDER BY
       o.object_id,
       i.index_id,
       ic.key_ordinal


The column key_ordinal in the sys.index_columns table is the order in which the columns are stored in the index. 

There isn't a key_ordinal column for the sys.stats_columns table. The column stats_column_id just replicates the index_column_id column of the object it references.

There is a slight difference in the wording of the article sys.stats_columns (Transact-SQL) for the column stats_column_id:


  1-based ordinal within set of stats columns.


...and in the article sys.index_columns (Transact-SQL) for the key_ordinal column:


  Ordinal (1-based) within set of key-columns.


I reckon that the index_column_id (sys.index_columns) and stats_column_id (sys.stats_columns) are the equivalent of each other and that only the sys.index_columns table has an ordering column, namely key_ordinal.