THE TESTUDO
The tortoise formation was one of the prime examples of Roman ingenuity at warfare. When deployed in such a way, the legionaries became virtually invulnerable to arrows or objects dropped from defensive walls.
THE WEDGE
The wedge was an aggressive formation used to 'crack open' enemy lines. Relatively small groups of legionaries could form such a triangle and then drive their way into the enemy ranks. As more Roman soldiers reinforced the wedge from behind, the enemy line could be forced apart. As breaking the enemy's formation was very often the key to winning a battle, the wedge formation was vitally important battlefield tactic of the Roman army.
REPEL CAVALRY
The order to repel cavalry by Roman army officers brought about a defensive formation, in which the front rank formed a tight wall of shields with their pila protruding to form a line of spearheads ahead of the wall. Undoubtedly it would be very hard to bring a horse to break into that formation. The most likely occurrence would be that it would come to a halt of its own will ahead of the spearheads. It was at that moment that horse and rider would be at their most vulnerable against the ranks behind the first line of infantry which would then hurl their spears at them. Given the short distance and the training legionaries received, it is likely such halted cavalry, frantically trying to turn their horses around to retreat, whilst colliding with horses following in the charge, would prove very easy targets.
If one further considers the likely possibility of archers being present, as is the case on the photo above, the effect of this formation could indeed be devastating
SKIRMISHING
The skirmishing formation is essentially the opposite to the closely packed line of battle used by legionaries. It is a widely spaced line. Every second man of the line has stepped forward a few paces, effectively doubling the amount of ranks. However, the gaps created by this formation are always overlapped by the next line to follow.
The roots of this formation are more than likely to be found with the velites, the lightly armed skirmishers who operated ahead of the main force in the early Roman army.
The wide spaces allow each soldier great mobility. Its possible uses were manyfold.
It would make an advance over difficult terrain much easier. It could allow for swift attacks with subsequent quick withdrawals. It would allow for any friendly units falling back to pass through the formation.
It also could be used by a victorious army sweeping over the battle field, killing all that was left in its way.
THE ORB
The orb was a defensive formation in the shape of a complete circle which could be taken by a unit which had either become detached from the army's main body and had become encircled by the enemy, or a formation which might be taken by any number of units if the greater army had fallen into disorder during a battle.
It can hence be seen as a formation representing a desperate 'last stand' by units of a collapsing army. But also it can be seen as a disciplined holding position by a unit which has been divided from the army's main body in battle and which is waiting for the main force to rejoin them.
In either case, it is not a formation one would like to find oneself in, as it obviously indicates that they are surrounded by the enemy.
Naturally any officers or archers would be positioned in the centre of the orb, as can be seen in the example above.
Cheers
Roc