The best way to do Remedy -> Jira is to use Jira's REST API to make changes in Jira.
In the other direction, it used to be a complete nightmare because Remedy simply didn't have any decent way for other applications to update it (At the time I was looking at it, the developers said that if they had something like that, no-one would use Remedy because they'd all write an interface just to avoid having to use it). I'm hoping that's changed and you have a way to do it.
If you can establish how to do remote stuff in Remedy, then you should be able to code simple post-functions or listeners in Jira to update Remedy
I did it many years ago, with SOAP rather than REST (REST is better, and SOAP is going away) and some unholy bodge to get around the inflexibility of Remedy at the time (although some of it was done by simply sending Jira emails to Remedy). Remedy was classed as a legacy system before we had to do too much though, which I was very grateful for.
Neither would be useful to you at the moment. You'll need to explore what your version of Remedy exposes to allow Jira to update it, and cehck https://docs.atlassian.com/jira/REST/latest/for how to talk to Jira.
Beware of the email though - it can go into a loop if you don't think it through and is open to users using it so they don't have to use Remedy...
The documentation is far better than anything I would write. I've pointed you at the docs you'll need to refer to for most of the calls, but I should have added https://developer.atlassian.com/display/JIRADEV/JIRA+REST+API+Tutorialsas well
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