Why did the scutum have a horizontal grip?

[M. Demetrius]

Well, I've said this before, and having used both vertical and horizontal grips, I find the palm down horizontal grip the least muscular effort.

The vertical grip requires constant muscle to hold the shield, as its weight naturally pulls agains the thumb side, so it requires effort to hold the shield, even when more or less relaxing the arm, and letting the shield "hang low". The horizontal grip only requires that the fingers stay curved around the handle, and most of the forearm muscles can relax, yet the shield still stays up.

If you have to carry it very far, horizontal wins every time. The angled grip described above might very well be the most ergonomic, but it would restrict movement of the shield without rotation, which doesn't work very well with a semi-cylindrical shield. Its purpose is to cover as much of the body as possible, with the least movement needed. Pulling the scutum close to the body provides a huge area of protection for the least amount of muscular effort. It just didn't happen by accident that they used this arrangement.

Anyway, there was probably less punching with the center of the shield than some like to think. By extending the shield in that way, it's no hard task for an enemy to reach around the shield and either pull it aside, or stab/slash beyond it. Striking with the lower shield edge still keeps the shield carrier further out of reach of the enemy.