As always with this kind of question it helps enormously if you provide more context than you have - impedances, bandwidths, etc.
If you connect the other half of each twisted pair to ground, as you've illustrated, you're effectively connecting your signal to ground via the cable's distributed capacitance - this is probably not what you want and, depending on the length of cable, may seriously reduce your measurement bandwidth.
You can minimise the capacitance of the pairs by having the second wire in the pair driven to the same voltage as your signal. In an ideal world this would come from a low impedance source such as an instrument's guard terminal. You won't have that in your setup so you're better off connecting the pairs together as effectively one wire, or just leaving them floating. Grounding them is probably your worst option.