The intersection has the appearance of lines, which would correspond to a planar structure in the 3-d phase space. This cannot be quite right, since again that would give crossing trajectories. But clearly the orbit is not space filling, i.e. three dimensional. Actually the structure is a "fractal" with nonintegral dimension, that is actually very close to two because of the strong contraction of volumes for the parameters used.
From the location of one dot on the Poincare section you can predict roughly where the next dot will appear. There is a great deal of predictability in the chaotic dynamics! One way of showing this is to plot a "one dimensional return map" of successive values of one of the coordinates. Lorenz used a slightly different version, which turns out to be preferable for this case.
The return map (filled in after some long iteration time) has the appearance of a function, and leads to the study of "iterated one dimensional maps". (Again, looking on a very fine scale we would find that the points do not strictly define a function - there is some "fuzziness" to the line.) A great deal of information about the Lorenz dynamics can be intuitively understood from the return map. For example the intersection of the "function" with the diagonal line corresponds to a fixed point; the fact that the magnitude of the slope of the function is greater than unity shows that the fixed point is unstable. Indeed we can associate the chaotic motion with the fact that the magnitude of the slope is greater than unity everywhere leading to the sensitive dependence on initial conditions.