A new sheet of spacetime is isolated and added to the existing sheet, thereby yielding a pair of co-existing sheets ofspacetimes, which are four-dimensional inversions of each other. The separation of the spacetimes by the special-relativistic event horizon compels an interpretation of the existence of a pair of symmetrical worlds (or universes) in nature. Further more, a flat two-dimensional intrinsic spacetime that underlies the flat four-dimensional spacetime in each universe is introduced. The four-dimensional spacetime is outward manifestation of the two-dimensional intrinsic spacetime, just as the Special Theory of Relativity (SR) on four-dimensional spacetime is mere outward manifestation of the intrinsic Special Theory of Relativity ($phi$SR) on two-dimensional intrinsic spacetime. A new set of diagrams in the two-world picture that involves relative rotation of the coordinates of the two-dimensional intrinsic spacetime is drawn and intrinsic Lorentz transformation derived from it. The Lorentz transformation in SR is then written directly from intrinsic Lorentz transformation in $phi$SR without any need to draw diagrams involving relative rotation of the coordinates of four-dimensional spacetime, as usually done until now. Indeed every result of SR can be written directly from the corresponding result of $phi$SR. The non-existence of the light cone concept in the two-world picture is shown and good prospect for making the Lorentz group SO(3,1) compact in the two-world picture is highlighted.