I can understand why the $x=ct$ line is needed as follows:
If $c \Delta t \geq \Delta x$ then the event can be causally related to your current position, so the event could not appear in reverse order for any observer; therefore that event must be in the absolute future.
Similarly, if $c \Delta t < \Delta x$ then the event cannot be causally related, and it is outside the absolute future — some observers might find the order of the event switched.
Hence clearly the $x = ct$ line plays a role in delineating absolute vs relative future.
What I don't understand is where the $x = -ct$ line comes in. What role does that play? Why are events in the absolute future also under the constraint that $\Delta x \geq -ct$?