As we have remarked before, any simple attack can be delivered as a riposte or counterriposte. Doing so only requires that the conditions required for the simple attack are present in the riposte's time period. Thus a straight thrust works if the line is open and unstable or opening. A disengage works against pressure in a line or against a closing line, or even to the opposite side of an unstable partly open (or partly closed) line. However, how do we use the counterdisengage? After all, a counterdisengage requires that the opponent initiate action, a circular attempt to take your blade.
That raises the obvious question - when in a parry-riposte exchange does the opponent ever do a circular action to try to take the blade? After all your attack come in in 4th, she parries 4th and ripostes 4th, and you parry 4th, etc., etc. All of those parries are lateral; not a circular action to be seen anywhere.
However, there are two parries that use circular motions:
(1) circular parry - picks up the opponent's disengage and returns it to the original line. For example, circular 6 (Sabre 3) takes the disengage into 4th and puts the attacking blade back in 6th (Sabre 3rd).
(2) change parry - picks up the opponent's straight thrust and transports it to a new line. For example, change parry 6 takes the straight thrust in 6th and puts the attacking blade in 4th.
Note that I have not used the term "counterparry" - I avoid "counterparry" for four reasons. First, people conflate counterparry with counterripostes and confuse everything by calling the parry against the counterriposte a counterparry. Second, counterparries have always been circular movements but exactly which circular movement has changed several times in the last 200 years. As I teach historical, classical, and modern fencing, sometimes to the same students, using a descriptive term that avoids the confusion makes some sense. Third, change parries and circular parries do tactically different things in terms of setting up the riposte, and different terms adds clarity. Finally, using the term "change parry" is consistent with the way the blade moves in the change of engagement and the change beat.
In doing this I accept one added bit of confusion. Remember that a circular parry is not actually circular. Completing the action as a circle adds a circular vector to the opponent's movement, facilitating a rapid roll off to escape the parry. The correct blade movement is a teardrop. This is so for both circular and change parries - there is no difference in the mechanics of blade movement, only in what we are using that blade movement to accomplish.
So we have these teardrop shaped circular movements that can be used by a defender to achieve several key objectives:
... capture a disengage (the circular parry)
... capture a straight thrust or cut (the change parry)
... transport the opponent's blade to a line other than the opponent's desired line (both circular parry and change parry)
... set up the riposte in your desired line rather than his (both circular parry and change parry)
... introduce an element of surprise (both circular parry and change parry)
... and when the movement ends in a 6 (3 for Sabre) move the opponent's blade to a line with reduced likelihood of a successful completion or remise (both circular parry and change parry)
This means that your riposte must be prepared to deal with the opponent who reacts with a circular motion as a parry. This is particularly true of the opponent habitually does circular parries on recovery from the attack to clear the lines - normally this is circle 6 in Foil and Epee and circle 3 in Sabre.
The answer is the counterdisengage. Some definitions of the counterdisengage describe it as a disengage followed by a second disengage to return to the line of the original disengage to hit. That is technically correct, but has two faults, one theoretical and one practical. The theoretical separation into two disengages loses the continuous circular flow, and makes it into a two tempo compound action, which it cannot be if it is a simple attack or riposte, and which it clearly is not in operation. Practically, describing it as two disengages makes it more complicated than it needs to be. In simple terms a countedisengage is a circular movement to return to the original line of attack and deceive an opponent's change parry, circular parry, change of engagement, change beat, or other circular attempt to take the blade. It is executed by following the opponent's circular movement progressively inward toward the target throughout the circle to hit as it is completing.
So, we now have a logical set of ripostes to deal with the most common responses:
... direct riposte if the opponent's lateral parry is defective or slow or if we wish to set up a second intention or a disengage counterriposte.
... disengage riposte if the opponent's lateral parry is effective or if we want to surprise the opponent or complicate his defensive decision making.
... counterdisengage riposte if the opponent habitually uses a circular parry on the recovery.