I don't care for matchy rhymey type names but I would want them to have a similar feel. For example I wouldn't do one classic name like William and then something super trendy like Jagger. But I also wouldn't do something matchy/cutesy like Aiden and Hayden. Search for just names you like and go from there
One of the ladies in my birth club is naming her twin girls Charlotte and Kathryn. I think those pair perfectly. Twins' names should match in style but not in sound... like my aunts Judy and Trudy. Yikes.
I know 6 year old twins named Adam and Aidan. My husband also has aunts named Marlene and Darlene. Don't care for that at all. I agree with the others. Find names similar in style but not matchy.
As a mom of twins I can tell you that you absolutely do not want overly matchy or, God forbid, rhymey names. The confusion would be overwhelming. Plus, if they are fraternal chances are good that they will be complete opposites, particularly if they are your first. There's a whole psychology of birth order that comes into play that I won't bore you with. Anyway, they will likely appreciate having names that reflect their individuality.
I do absolutely agree with all the previous posters that completely different styles of name would be weird, though.
Re: Naming twins
My twins are boy girl, they do not match at all (ick) but are both very Italian so they go well together I guess.
Pick 2 names that you love. They are individuals and should be named as such not as a unit.