Non-interchangeability of time-like intervals I am reading Landau's Volume 2 of the course of theoretical physics. I have a doubt after reading the first few pages of it which I explain below.
Landau first defines intervals and on pages 5 and 6 shows that two events having time like interval between them can never occur simultaneously in any reference system. Then he goes on to construct a 2D space-time graph (for visualization) with an event O occurring at (0,0,0,0). Then he considers any event which occurs in future in that frame and is time-like w.r.t. origin and says on page no. 7, 

But two events which are separated by a time-like interval cannot
  occur simultaneously in any reference system. Consequently, it is
  impossible to find a reference frame in which any of the events in
  region aOc occurred "before" the event O, i.e. at time t<0.

The argument above just proves that because interval square should be positive, i.e. the events can't be simultaneous. But, if I replace the difference in time in the original frame with its negative in my proposed frame and let the space distance between them to be same in both frames, then I get an in my proposed frame an interval which is time like but in it the order of events is changed. Am I making some gross error or Landau has missed some argument?
 A: If you believe that (a) timelike separated events cannot be simultaneous in any reference frame, and (b) the set of inertial frames is (in some appropriate sense) a continuous set, then L&L's conclusion follows. After all, if there were two frames in which the order of two timelike separated events differed, then by continuously transforming one frame into another, you could find one in which they were simultaneous.
But without some such additional assumption, you're right that the conclusion doesn't logically follow. There are coordinate systems that preserve the spacetime interval but flip the direction of time, such as the substitution $t\to -t$ that you mention. As BebopbutUnsteady observes in a comment to Karsus Ren's answer, we often use the term "orthochronous Lorentz transformation" to refer to a transformation that preserves the direction of time. The full group of Lorentz transformations (i.e., of all transformations that preserves the interval) includes both orthochronous and non-orthochronous components, which are not connected to each other. Physically, we usually only consider the orthochronous ones. 
You do have to be careful with the terminology: sometimes people use "Lorentz transformation" to mean just the orthochronous ones; sometimes it's the full group.
By the way, pretty much the same thing applies to spatial reflections: is $x\to -x$ (leaving $y,z,t$ unchanged) a Lorentz transformation? After all, it preserves the spacetime interval. Often we refer to non-reflecting Lorentz transformations as "proper." So when people are being careful with their terminology they often refer to "proper orthochronous Lorentz transformations."
A: I don't know what Landau says, but space-time interval doesn't distinguish order. To derive the non-interchangeability of time-like intervals, Lorentz transformation must be explicitly used.
