I believe when they say reconstrucked Cloud that they don't really mean rebuilt him as you would a robot or something. It's just how they worded it. Cloud wasn't really rebuilt, but more or less enhanced by the Jenova Genes. When they say rebuilt they mean they turned him into what they wanted him to be. Just like any other military, they break you down mentaly, and rebuild you the way they want. Not just in form or shap, but more or less mentaly. So yes in a way he was rebuilt, but not litteraly in a sence. <--- Getting what I am saying so far? He's still Cloud just more inhanced. That is all. Think of when after Tifa helps him find his true self again. It was just a mental thing more or less. They only fisical change was his body become stronger because of the Jenova Genes.