I'm trying to do some research into these types of logos but I can't find anything of use on Google using the buzzwords - 'Cross logo', 'X logo', 'X with text around logo'. Is there a certain name for this design?
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There's not really a formal name but you can find it as Crossed X logo. Or just Crossed logo but that will yield more diverse results, namely including those with religious crosses as well.
Nowadays its frequently associated with "hipster branding," as evident by these resources:
It's a pig pen cipher used for encryption/decryption. It's on old coins. This is nothing new, and it's not from some NY hXc... They aren't OG.
I think the well known railroad crossing sign (R X R) cannot be ignored here. Many places which employ this logotype, as mentioned above, only have letters on the sides. Some also include an enclosing circle.
The RR crossing symbol is something that's in the public consciousness, but people don't probably think of it immediately when they see a logo for a bar or beard wax store or leather goods shop.
The sign dates to the 1920s/1930s. The design was standardized in the Manual on Uniform Traffic Control Devices, (MUTCD) adopted in 1935.