For example, in the US:
In 1834, the United States fixed the price of gold at $20.67 per ounce, where it remained until 1933. (EconLib)
Does this mean that anyone with US\$20.67 could walk up to any bank in the US and demand an ounce of gold? Conversely, could anyone with an ounce of gold demand US\$20.67 in paper money from any bank?
Or were the rules of redemption more restrictive than that?
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
EDIT
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.
Under the gold and silver (bimetallic) standard I have read that units other than banks, such as governments and firms, may have also been able to issue paper notes which promise redemption for specie at the pay window of the issuer. However this created credit default risk or the risk of non-payment upon presentation of the note. Therefore the paper notes would trade at a discount in commerce based on efforts to determine the creditworthiness of the issuer.
Brief History of the Gold Standard in the United States
https://fas.org/sgp/crs/misc/R41887.pdf
Various banks conducted much of their business based on the issuance of notes. Taking deposits and making loans, the banks needed only a fraction of their total assets held as coin on hand. The rest could be held in the form of interest-earning loans, and issued as notes promising to pay the bearer on demand an amount of gold or silver on presentation of the notes. The notes were not legal tender, but circulated on the strength of the promise to redeem. Sometimes the notes passed at a discount that represented the possibility that they would be dishonored. And the discount varied with the distance from the bank and its reputation for soundness. The congressionally chartered First and Second Bank[s] of the United States were able to issue such notes on a national scale through branches throughout the country. These notes were not legal tender, but tended to pass at par with no discount (i.e., at face value).16 By presenting for redemption the notes of state-chartered banks that it received from customers, the Bank of the United States was able to help state bank notes remain at par as well.17Banks were not always able to keep their promise to redeem notes, however, even when the banks were solvent. When unusually large numbers of customers presented notes for redemption, the demand for gold and silver exceeded what the banks had on hand. Periodic financial crises led to suspension of convertibility of notes. In such periods, paper money and metallic money diverged in value, and one was no longer a perfect substitute for the other.
