For some reason Sony decided to ship the Xperia ZL (C650X) with display calibration software but left the Xperia Z out (C660X). The software allows users to separately adjust white balance, colour saturation and hue. Given the criticisms in some quarters over the Xperia Z display, it is surprising that Sony didn’t include the software, although we may see it appear in an upcoming update.
Thankfully the apk has been ported over from the Xperia ZL if you don’t want to wait. Your Xperia Z will need to be rooted for the calibration app to work, as you need to set the right permissions. Instructions to get this working can be found here.
Thanks Anirudh and Luke!