Only way I can figure out how to get this information is from the SOAP API.
It seems like the only thing in the entire SharePoint object model that changes when I update that field is
AnonymousPermMask from the Lists.asmx service.
This does not appear to be available from any of the apis except the Soap API? Why isn't this available? I created a uservoice about this:
AnonymousPermMask="68719546465" - When checked.
AnonymousPermMask="0" - When not checked.
Request to get it:
Content-Type: text/xml; charset=UTF-8
- Request body: