Does the Table order in the ON clause matter in LEFT and RIGHT JOINs ?

Basically, are the following two Statement functionally identical or not ?

SELECT <fields>
FROM TableOne a
	LEFT OUTER JOIN TableTwo b ON a.JoinColumn = b.JoinColumn

SELECT <fields>
FROM TableOne a
	LEFT OUTER JOIN TableTwo b ON b.JoinColumn = a.JoinColumn

I tend to use the first version but, since I sometimes need a "AND b.OtherColumn = 'XXX'" clause, it seems more consistent to use the second version so that the table order matches.

I'm pretty sure that I qualify as a SQL expert with 30+ years experience and I am pretty sure that the ON clause order does NOT matter, but I thought I should double check.