a) The Society has not defined a formation called "Diagonal Rights& Lefts" - but the Irish Rover has been danced for more than 35 years, and people have been calling the last 8 bars "Diagonal Rights& Lefts" for at least that long.
b) The problem is that the deviser, J Cosh, did not call it that - he just said who changed hand with whom, and with which hand.
c) So when the Society published "The Whistling Wind" it used a similar form of words for the last 8 bars (although this has not prevented people describing the last 8 bars as "Diagonal Rights & Lefts"!).
To answer the specific question:
The difference between the first 4 bars of "Rights & Lefts", and the 4 bars of "Half Rights & Lefts" is the "polite turn" at the end of the second.
So Diagonal Rights & Lefts (or Rights & Lefts on the Diagonal) would last 8 bars, be danced by the same 4 people, and there would only be a "polite turn" at the very end.(The most recent dance published by the Society which contains this is "The Dream Catcher", but that is in a square set).
Therefore the formation which is normally danced (i.e.the last 8 bars of both "The Irish Rover" and "Whistling Wind")strictly speaking consists of Two, "Half Diagonal Rights & Lefts", as there is a "polite turn" at the end of both bars 4 and 8. It does occur as a 4 bar formation in "Eileen Watt's Strathspey".