Callum Simpson,   Yes I can confirm that the E5645, as well as the E5620, are both compatible in both servers, and also have the same part #.   Let me know if this helps.  
Long story short I’ve got 2 servers one being a power edge r610 with 6 cores and the second server is a power edge r710 with 4 cores if I want to take the CPUs out of my power edge r610 will they be compatible and work in the power edge r710 specs currently  R610 cpu: Intel(R) Xeon(R) CPU E5645 @ 2.40GHz r710 cpu: Intel(R) Xeon(R) CPU E5620 @ 2.40GHz  
Are R610 and R710 CPU the same?