Motivation for preservation of spacetime volume by Lorentz transformation?

Motivation for preservation of spacetime volume by Lorentz transformation?

My favorite way of deriving the Lorentz transformation is to start from
symmetry principles (an approach originated in Ignatowsky 1911; cf. Pal
2003), and one of my steps is to prove a lemma stating that the Lorentz
transformation has to preserve volume in spacetime, i.e., in fancy
language, it has to have a Jacobian determinant of 1. My visual proof of
the lemma (for 1+1 dimensions) is given at the link above, in a figure and
its caption.
Intuitively, the idea behind the proof is that it would be goofy if a
boost could, say, double the area, since then what would a boost in the
opposite direction do? If it's going to undo the first boost, it has to
halve the area, but then we'd be violating parity symmetry. The actual
proof is actually a little more complicated than this, though. If a
rigorous proof could be this simple, I'd actually be satisfied and say the
result was so simple and obvious that we should call it a day. But in fact
if you want to get it right it becomes a little more involved, as shown at
the link where I give the actual proof. One way to see that the intuitive
argument I've given here above doesn't quite suffice is that it seems to
require that area in the $x$-$y$ plane be preserved under a boost in the
$x$ direction, and that's not true.
This is different from the approach based on Einstein's 1905
axiomatization. In that approach, you derive the Lorentz transformation
and then prove the unit Jacobian as an afterthought.
Although my proof of the unit Jacobian from symmetry principles works,
I've always felt that there must be some deeper reason or better physical
interpretation of this fact. Is there?
Takeuchi 2010 says on p. 92:
This conservation of spacetime area maintains the symmetry between [...]
frames, since each is moving at the exact same speed when observed from
the other frame, and ensures that the correspondence between the points on
the two diagrams is one-to-one."
This seems clearly wrong to me, since you can have one-to-one functions
that don't preserve area. (Takeuchi is writing for an audience of liberal
arts students.)
Mermin has a very geometrical pedagogy that he's honed over the years for
a similar audience, and he interprets space-time intervals as areas of
"light rectangles" (Mermin 1998). Since he uses the 1905 Einstein
axiomatization, conservation of area, which is equivalent to conservation
of spacetime intervals, comes as an afterthought.
One thing that bugs me is that the area-preserving property holds for
Galilean relativity (and my proof of it holds in the Galilean case without
modification). So any interpretation, such as Mermin's light rectangles,
that appeals specifically to something about SR seems unsatisfying. The
notion of area here is really just the affine one, not the metrical one.
(In Galilean relativity we don't even have a metric.)
Another way of getting at this is that if you start with a square in the
$x$-$t$ plane and apply a Lorentz transformation, it becomes a
parallelogram, and the factors by which the two diagonals change are the
forward and backward Doppler shifts. Conservation of area then follows
from the fact that these Doppler shifts must be inverses of one another.
Laurent 2012 is an unusual coordinate-free presentation of SR. He
interprets $\epsilon_{abcd}U^aB^bC^cD^d$ as a 3-volume measured by
observer whose normalized velocity vector is $U$. Restricting to 1+1
dimensions, $\epsilon_{ab}U^aB^b$ is the length of vector $B$ according to
this observer. He gives the example of associating affine volume with the
number of radioactive decays in that volume. The implications of this are
vague to me.
What is the best way of interpreting the preservation of spacetime volume
by Lorentz transformation?
Please do not reply with answers that start from the known form of the
Lorentz transformation and calculate the Jacobian determinant to be 1. I
know how to do that, and it's not what I'm interested in.
W.v. Ignatowsky, Phys. Zeits. 11 (1911) 972
Bertel Laurent, Introduction to spacetime: a first course on relativity
Mermin, "Space-time intervals as light rectangles," Amer. J. Phys. 66
(1998), no. 12, 1077; the ideas can be found at links from
http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html , esp.
http://www.ccmr.cornell.edu/~mermin/homepage/minkowski.pdf
Palash B. Pal, "Nothing but Relativity,"
http://arxiv.org/abs/physics/0302045v1
Takeuchi, An Illustrated Guide to Relativity