I was unable to get my screen to rotate when I changed to tablet mode and have spent almost a day trying to fix this. As I finally found a solution I thought I would post it here to hopefully be of use to someone.
@HP: there are various flavours of this solution out there, but none of them worked for me. You should be ashamed of yourselves that you have apparently not sufficiently addressed an issue that has been troubling people since 2015!!!
Issue: Flip screen to tablet mode, then rotate 90 degrees. Screen does not rotate
Solution: (tl;dr: update/downgrade rotation sensor firmware to Dec 8 2015 version (sp74158.exe) )
Start using your laptop in laptop mode (i.e. with keyboard working)
1) in your search (at the very bottom left of your screen is a Windows icon. click the circle to the right of this) type 'regedit' and hit enter. Allow the app to make changes when prompted. This opens up the registry editor.
2) In the left frame of the registry editor expand the entry 'HKEY_LOCAL_MACHINE' and then \SOFTWARE\Microsoft\Windows\CurrentVersion\AutoRotation --> The value of 'SensorPresent' should be 1. If it isn't, change it by double clicking and entering a 1 instead of the 0. --> The value of 'SlateEnable' should also be a 1; change this too if it is not. If you do not see 'SlateEnable', don't panic. I didn't at first either...just follow the below and check at the end if you maybe need to change it.
[Other answers in this forum propose changing 'Enable' to 1 as well. This will not hurt, but I think it might not be necessary - once you have a correctly working version of the firmware for the sensor installed this value will dynamically be updated.]
3) Now google 'HP Spectre x360 Software and Drivers' and click the link to the support.hp.com page. Enter your Product or Serial number (case sensitive!!) and check that you have the correct laptop on your screen (incorrectly types serial number will still turn up a machine, but the wrong one - check well!). Select the 'Software and Drivers' box and click the GO button to take you to the 'software and driver results'.
4) Under Driver-Chipset you should see a driver for the Sensor: 'Intel Integrated Sensor Solution Driver' with date May 27 2016. You could try using this file, but it refused to work for me. I chose the previous version: Dec 8 2015. Download and run this file (sp74158.exe), close all other open programs and let it do its magic
5) Reboot your computer.
6) Now flip the screen over and rotate by 90 degrees. If all went well you should see it rotate. If this doesn't happen, click on the notification icon (rightmost icon in your taskbar) and click on 'Rotation lock' and set this to OFF.
--> A number of responses have mentioned that the rotation lock button is greyed out. However, it seems to come to life in tablet mode. --> This is the same for button 'Lock rotation of this display' in the Display settings: greyed out in laptop mode, but accessible in tablet mode. I think this is why I am unable to rotate my screen using crtl-alt + arrow keys...
If things are still not working as expected, redo steps 1 and 2 above and make sure that 'SlateEnable' in the registry is set to 1.
I was so frustrated with the trouble it took me to get this working that I registered with HP only to be able to post this. I really hope it helps some of you out there!
<frustration> @HP: guys, the Spectre X360 was supposed to have been developed in close conjunction with Microsoft and Intel. Really not cool that this issue has been unaddressed for so long. When do you think we can see the ctrl+alt + arrow issue fixed? Or the greyed out rotation lock in tablet mode? Or that the latest version of the firmware actually works better than the previous? </frustration>