This is an English translation of Part 6 of a seven-part article by Nakamura Sensei, 7th Dan. The link to the original article in Japanese is: http://www.geocities.jp/gazo_asidachi/kumagro_niki/iyashino_kendo/iyashi_01.htm. Posted with permission and translated by KendoNotes.
Well then, with only men-uchi strikes, the opponent would definitely exploit this. For the 7th Dan level, drawing out the opponent and striking a yuko-datosu is desired.
If the opponent raises the hands to strike in response to the shift of one’s center of balance starting with one’s right leg to execute men-uchi, then it is kote, nuki-do or, if delayed with respect to the opponent’s men, kaeshi-do. With a seme that moves one’s center of balance starting with the right foot, since the right hand is not raised, one can switch to a variety of waza.
This time, I shall write about kaeshi-do. Although there many ways of striking, if you would like to use it (the kaeshi-do), please use it.
After mastering how to strike this, my fear of men has disappeared. No matter how fast a men, I simply counter (with a kaeshi-do). However, if I use this too much, the opponent stops striking men and I am in trouble. Think of using this technique in moderation.
Here, many ways of inviting the opponent in (sasoi) are available. For example, from the position of kamae, invite the opponent without moving either foot, slowly shifting one’s weight forward and backward between the width of the back foot and forward foot. The head moves slowly forward and returns. Then if one advances the men slightly faster and slightly more, the opponent is invited and comes in strike men. The feeling at the time of invitation is, as expected, like giving the men.
As the opponent comes in to strike men, counter at the instant the men would be contacted. At the instant the opponent thinks he or she has the men, respond with a kaeshi-do.
Now then, on to how to strike do. It is similar to the 7th kata in kendo kata. Step the right foot slightly towards the right and, at the same time, raise the hands to receive the opponent’s shinai. At the instant it is received, send the left foot diagonally towards the right and strike do. To come out, escape using ayumi-ashi, immediately turn around and adopt zanshin. At the instant of hitting do, please have sufficient composure and calmness to check to see where your shinai strikes the opponent.
With regards to advancing the left foot, it is advanced at the same time the do is struck. Although this can be done slowly as the legs cannot move that fast, given a slow men, this (going slowly) is possible. However, given a fast men, this does not apply. Note that the image of advancing the left foot and striking is important.
In practice, when responding with a kaeshi-do in response to a fast men and when intending to advance with the left foot, the left hip first rotates towards the right. At the time of the hip rotation, the do is struck. Of course, afterwards, the left foot is pulled in by the hips and one advances forward. At that point, the do-uchi has already been completed and the body is escaping out (of the strike).
Some additional comments. On the first invitation (sasoi), there is no rule saying that it must be with this. In the middle of koubou (attacking and defending), once one can do kaeshi-do, composure and calmness develops. Because this may be so easy, rather than being content with this, please do not forget about seme in your kendo. I wish you excellence and progress in this.
 Translation note: The text was translated literally here. I believe it refers to no hardfast rule that one must succeed or strike with kaeshi-do on the first invitation.