You could. The only ones that you'd need to test for rectally are gonorrhea and chlamydia. You might ask your doctor for an anal pap smear, as well, to look for anal HPV. Did you also have unprotected vaginal sex with this person? If you did, and didn't get anything from them vaginally, it's not likely that you got anything anally, but it wouldn't hurt to check. How long has it been since you were with this person? If it's been less than a few years, you could have it and not have symptoms.
The others - HIV, syphilis and herpes - would show up on a blood test no matter how you were infected.