If possible, I would like to be able to still use legacy Remedy for ticket flow with our organisation worldwide but for our nationwide IT support JSD seems to be way more efficient and flexible. So being able to automatically open tickets incoming from Remedy in JSD and back would be great, favorably keeping the same ticket name/number.
Mmm, well, you won't be able to keep IDs the same, unless you can coax Remedy into using Jira's issue ID scheme. Of course, the easy solution there is a field in Jira to keep the Remedy ID and a field in Remedy to hold the Jira one. You'll need to rig up something to duplicate the bits of data from each system in the other. Jira has several ways to handle integrations you could look into (web hooks, post-functions, email, and most importantly, the REST interface), but the choice of which to use depends mostly on how you're going to approach it. From memory, Remedy is about as friendly as GrumpyCat when it comes to good ways to talk to other systems, so I'd start with thinking about what that can do, and worry about Jira once you've decided how to script/write that side of it.
The standard docs are at https://docs.atlassian.com/jira/REST/latest/ But I find https://developer.atlassian.com/display/JIRADEV/JIRA+REST+API+Tutorials is generally more useful as a starting point. I think your minimum requirement is probably met with "A Jira custom field that refers back to the Remedy ID" (although I'd like to do it in reverse as well as get that basic link)
...be more productive while being fun to use at the same time. For some, getting started can be a bit intimidating. This is especially true if Jira Service Desk is your first exposure to Atlassian...
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