Many types of fiber connections, including 1000base-LH/LX, use two strands of fiber. However the strands are actually used individually as dedicated TX and RX on each and the traffic on each strand is entirely unidirectional.
So when you are connecting two such devices, you need to cross the strands at some point so that TX on one side matches up with RX on the other. If you do not do so, you end up in a situation where TX on one side is connected to TX on the other and RX is connected to RX. This results in no link connectivity.
Fiber patch cords typically come from the manufacturer either straight through or crossed depending on the manufacturer. Depending on the number of cross connects, either one could give you the end result of a straight through connection (i.e. an even number of crossed connections results in a straight through connection).
In a cross connect situation like you describe, where there is connectivity between floors by a third party, often they will provide you a straight through connection. The reason for this is that you are free to use the cross connect with any fiber solution you wish, which may include a solution that uses a single strand bidirectionally. However there may be a number of different cross connections along the path, and if they don't tell you specifically whether it is straight through or crossed, it may be either.
Most fiber patch cords use some sort of plastic piece to connect the two strands of fiber together for easier insertion and removal. There is usually some means of detaching the strands from this plastic piece so that you reverse the strands and re-attach, but the exact means varies depending on the manufacturer (and sometimes the production run).
For example, some of these will open on some sort of plastic "hinge" that will open and allow the strand to be removed. Others may require a "twisting" motion to remove and insert. I know I have come across GBIC based cables where there are two plastic halves that enclose the strands, but I haven't seen that personally with LC connectors but I would imagine it's possible as well.
