Large Transport; Low Wing; 2 Turbojet Eng
Situational Awareness; Confusion; Communication Breakdown
Conflict Ground Conflict; Critical; Deviation / Discrepancy - Procedural Clearance
Flight Crew Took Evasive Action
Procedure; Environment - Non Weather Related; Human Factors
We had pushed back off gate at MCO and were starting engines.  I heard company aircraft check in with Ground Control taxiing in.  Ground told them to taxi 'but you'll probably have to wait for company (us) to taxi before you can proceed to the gate.'  Engines stabilized; I called Ground for taxi and was instructed to taxi out (no reference was made to company traffic).  Captain added power to taxi; but as we cleared right we noticed company aircraft passing close abeam and then cutting in front of us on their way to the gate. The Captain pulled power to idle and held brakes. Had we continued I believe there would have been a ground collision; especially when we turned and swept tail.  There wasn't enough room for company aircraft to maneuver to taxi in on the taxi in line; so he was offset as he parked.  I asked Ground Control the callsign of company; and tried to contact them on the OPS frequency but they didn't answer.  Once they were clear of us and parked we proceeded to taxi out.
After a normal push back; straight back of gate at MCO; we were in the process of starting the engines and received a salute and acknowledged it. We switch to Ground Control on 121.8 and heard inbound being told by Ground Control to enter the Ramp and that it looked like he needed to wait until the pushed backed aircraft (us)  moved before he could taxi to the gate (the same gate we had pushed from and were looking straight down the lead in line). After starting we were cleared to taxi to 18L; and via ramp exit then E; C; B1; 18L. We checked clear left and right and as we just started to taxi when we saw an aircraft coming on our right at a fairly fast speed for the ramp between us and the terminal building and gates on our right side so we stop immediately as we could not see if his wing would make it by us. Much to our amazement he keep coming with not enough room to fully swing in front of us he turned back in to gate not even on the lead in line and then even more amazing at that point the marshallers gave him guidance to park in; on an angle not even on the lead in line. So not only did this crew put my aircraft and his in a dangerous situation for a ground incursion; the ground crew did not stop an unsafe situation on the ramp.
A Large Transport was cleared to taxi out of a MCO Ramp while another large transport was cleared into the same Ramp. But after ATC said the inbound would probably have to wait; they didn't; which created a dangerous conflict.