Yes, you can still go for STI testing during menstruation.
HIV and syphilis testing is from a blood sample and would not be affected at all.
For STI like gonorrhea and chlamydia they can check for them with a pee test or a swab from the vagina. Even doing a swab test for these STI while you’re having your period is no problem. Some tests can be affected like a PAP test when your having your period but standard STI/HIV tests are no problem.
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