〜つ〜つ form means
〜たり〜たりしながら. It is followed by how or why those actions occurred. The two verbs are usually "opposite" or "contradictory" actions. I think it is a more formal construct, but not necessarily archaic.
Here are some example sentences from my book:
- マラソンの最後の500メートルで二人の選手は抜きつ抜かれつの接戦になった → The last 500 meters of the marathon become a close battle between the two runners as they kept trading out for first place.
- 風に吹き飛ばされた赤い帽子は木【こ】（こ）の葉のように浮きつ沈みつ川を流れて行った → A red hat that was blown about by the wind kept rising and falling (sinking) like a (tree) leaf as it went on down the river.
- 変な男の人がうちの前を行【ゆ】（ゆ）きつ戻りつしている。何をしているんだろう。 → There is a strange man going back and forth (going and coming) in front of our house. I wonder what the heck he's doing!?
As far as your example
持ちつ持たれつ, a whole sentence and/or more context would be easier to really decipher the meaning.