This is one of the conundrums of this war. I would expect, initially, that all militia and regular cavalry were issued with swords. Given the numbers produced, I cannot see why most new cavalrymen were not. Cavalry, however, are not the only 'mounted troops' as this also includes horse artillery (gun teams attached to cavalry brigades) and 'mounted infantry' - which was a relatively new revival of the dragoon whose principle weapon was the rifle, not the carbine or sword. The cavalry were expected to fight from horseback, the others were not - although many did.
Cavalry were used as scouts and, in number, as shock-troops to scatter disorganised bodies of enemy, particularly after a heavy round of volleys or salvoes - to finish off any chance or reorganisation. Outside the main field armies, there was little need for them to be used in this role, so many were treated as 'mounted infantry', no matter what they were armed with, since they were unlikely to be used in the traditional role of cavalry other than patrolling or reconaissance.