These are the specific permission toggles:

Additional Options Permissions Add Agile Objects
Additional Options Permissions Delete Agile Objects
Additional Options Permissions Manage Agile Objects(Admin)

I have found this bug [JIRAALIGN-1030] which referenced these three files:
https://jira.atlassian.com/secure/attachment/391089/391089_add+agile+object+toggle+investigation.txt
https://jira.atlassian.com/secure/attachment/390934/390934_Delete+agile+objects+toggle+report.txt
https://jira.atlassian.com/secure/attachment/391140/391140_manage+agile+objects%28admin%29.txt

It would be great if they could be documented in more detail than they currently are, as these three toggles seem to have wide-ranging impact, and yet they are hidden away in Additional Options.