Yes, it is absolutely possible to move around both the modes of LTE. Below is the Image where all the Inter-RAT Communications are possible.
This is Redirection, Reselection, Handovers and other Inter-RAT combinations in LTE
Interoperability between LTE FDD/TDD network:
In countries where FDD and TDD are both in use, it would be interesting to see Dual-mode LTE terminals that would support both TDD and FDD and it should be possible to do a reselection as well has handovers from one mode to another.
It's the task of Radio Resource Management from both sides to initiate the Handover command. It comes under E-UTRAN Handovers for details see section 5 of spec 36.133
Here you can see the requirement for support of radio resource management.
Now the thing is either it will be connected state or in idle state. If it will be in idle state then reselection will happen but if it is in RRC_CONNECTED state then handover will be initiated.
For the whole details i have found a good doc which is actually in use for testing purpose but will give you a better idea about this scenario.
For mobility in Idle mode and connected mode please refer this specification
See here: Radio Resource Management (RRM) conformance testing
Section 4 is E-UTRAN RRC_IDLE State Mobility and section 5 is E-UTRAN RRC_CONNECTED State Mobility.
For complete knowledge read 36.410 (S1 Procedures), 36.420 (X2 Procedures) to understand the mobility completely.