Steps to reproduce:

1. Create an issue.
2. Search "created > TODAY". (where TODAY = today's date)
3. JIRA will return the created issue.
4. Make the issue due today.
5. Search "due > TODAY".
6. JIRA will not return the relevant issue unless query is "due >= TODAY".

I expect that the second behaviour is the more logical one. Note that the Created and Updated fields appear to have the same problem.