Fergus,

Both the Bishop and the Rook do indeed have orthogonal lines of movement.
I touched on this this in a 12-13(?) comment directed to Charles 
concerning why Rooks, and not Bishops, are usually described as are
orthogonal movers; basically, my answer was that its a convention --
meaning a tradition -- and a bow to common usage; since Bishops are
described as diagonal movers it seems relatively harmless to describe
Rooks as orthogonal movers. In fact Solomon Golomb (who developed
Cheskers, Pentominoes and was a leading light in recreational math), in a
write-up on Cheskers, once described Bishops as Rooks on the 32-space
board formed by one color of the chessboard, and Camels (Cooks in
'Cheskers') as Knights on the same board. 

I certainly don't find it a problem to think of Bishops as orthogonal
movers, and I think any rule that uniquely identifies Rooks and not
Bishops with 'the possible set of orthogonal movement patterns' would be
somewhat deficient, since they are simply rotations. 
[Aside: I have used the 'Cheskers' game as an inspiration for a very odd
game called 'Dichotomy Chess' (modest - goal variant), where I also
tacked on a Dabbaba-rider + Ferz (B+K on 32!)].  

My comment about 'straight lines'? It illustrates a construction
guideline that does give rise to straight lines in one context (planes)
and arcs in other (spheres), even though we might have been trying for
'meaning the same thing' and used a rule that is used to produce
straight lines in planes. I certainly don't consider straight lines and
arcs the same thing -- and I don't feel a need to call them both straight
lines, or both arcs. They are simply analogous with respect to the rule of
construction, but do not fully represent the same meaning. 

Walking the 'straight-lines' over to the orthogonal discussion: a rule
that does produce paths of orthogonal movement on a square-grid and can be
applied to produce paths on a hex-grid does not replicate orthogonal
movement on the hex-grid -- it produces sets of movement paths through a
point that are orthogonal on square-grids boards, but not on hex-grids.
Analogous with respect to the rule of construction (and even using the
word right angle -- so it must be legit?) if we apply the rule to square-
and hex-grids, but producing results not reflecting the same type of
thing. 

On a hex-grid, the simplest orthogonal movement pattern involves an
'edge-path' and a 'point-path' (e.g., vertically and horizontally on
the Glinski board). A while ago (few weeks), I indicated to Charles G.
that this is a mapping of a standard Bishop (e.g., from a chessboard
rotated 45 degrees) that was 'halfbound' as opposed to the
'thirdbound' pattern of g-Bishops. 

To try and wrap up my end of this discussion of 'angles dashing from a
hex in a plane'. There exists a usage convention (tradition with a group
of supporters) for using 'orthogonal' and 'diagonal' to describe some
possibly paths on a hex grid. The usage (1) isn't especially apt, since
it conflicts in some important ways with the usual meaning of orthogonal
and diagonal in both chess and mathematics (especially plane geometry) and
(2) suggests a 'rightness' (based on the analogy to standard chess) that
is misguided, a frequent source of confusion, and somewhat stifling for
developing other approaches to hex chess. I therefore feel its a usage
ripe for replacement.