Surveyors take sets of measurements in at least two different ways in order to work to some extent in a self-checking manner. A common example is a 'face left' half set of observations followed by a 'face right' half set, together making a 'full set' of observations. This face left and face right approach involved turning the telescope 180 degrees vertically between half sets.
This example here seems to be a way of spotting big (human) errors in reading the readings by taking two half sets of observations and shifting the horizontal circle (on which the angle readings are read) by a random amount between half sets. I have never used it nor have I seen any previous reference to it. Hence my medium confidence.
I would imagine there are many different variations of half sets around, depending on what is being observed and checked for.