Why does a caloric reflex test induce sideways movement of the eyes?
The horizontal (= lateral) semicircular canal isn't truly horizontal.  Have a look at this lovely image from Wikipedia, which gives a nice explanation.  If you keep track of the helicotrema of the cochlea there, which is on the lateral aspect, you can see that the horizontal semicircular canal connects with the utricle higher laterally than medially.
This is somewhat confirmed in Maxwell, 2016 Figure 1 which illustrates that the &quot;horizontal&quot; semicircular canal is only horizontal when the eyes are lowered by 30 degrees from a standard anatomical position.  (Bearing in mind that the two lateral semicircular canals are not coplanar, but diverge by 15 degrees).  That paper also confirms that buoyancy is a key mechanism in the caloric test.
Because the lateral end of the horizontal canal is higher, treatment with cold water will draw the endolymph away from it to the medial end, and hence away from the ampulla (&quot;ampullofugal lymphokinesis&quot;).  The patient would feel they were turning laterally relative to the the position of the fluid (taken to be fixed in space).  This leads to two types of eye motion - the vestibulo-ocular reflex to maintain fixation by moving the eyes medially, and saccades to follow the position of the &quot;moving&quot; stationary object that the patient is trying to focus on by moving the eyes rapidly laterally.