It depended on the year and if a banking suspension was in effect in the town. In 1834 it was absolutely true that you could walk up to a bank and demand one ounce of gold or in some time periods silver for \$20.67. Of course, you would have to accumulate \$20.67 which was a lot of money. The Treasury would demand upon receipt of a banknote any and all gold represented by those notes at all times during this period.
You should double check the amount because I thought it was pegged at $20 over that time period.
Temin, Peter. The Jacksonian economy Norton New York 1969
The converse was not true unless it was a minted coin. The period was known as the free banking era. I hadn't noticed the converse question. You should read Temin's book. There are also other books on the free banking era. Paper money was not printed by the United States until Lincoln authorized the creation of demand notes under an 1861 Act. See
Statutes at Large. 37th Congress, 1st Session. Chapter XLVI.