It is in fact possible to power the Pi via PoE using a PoE Splitter: £9.90 inc VAT from Amazon for the GB speed model. The data is piped through the Ethernet cable on the device and power from the switch is diverted to the MicroUSB cable attached to it which powers the Pi. So yes, can power the Pi via PoE, but via the standard Power socket on the Pi. You can also of course use a PoE Hat but if you need to use another hat, things start getting ugly ;-)
The PoE Splitter is tested and known to work powering both a Pi and even a 7" LCD monitor as in below picture:
Picture of the setup working correctly from the front: 
The POE switch I use with my the POE Splitter is a ZyXEL_GS1900-8HP-GB0102F, currently £88.35 ex-VAT from Amazon. By enabling and disabling the port the POE-powered Pi is connected to I can remotely start a Pi which is in a down-state (either not booted or shutdown using sudo systemctl poweroff ). This is a really great switch with a lot of bells & whistles for a very reasonable price. Even has SNMP and IPv6 support.
