A bidirectional transformation applies to all observers, and so must work for any direction, including observer and observed with the roles switched. A physics for all observers should be bidirectional if possible. This works for mechanics but for thermodynamics entropy is inherently directional.
The Galilean transformation is for one direction with no characteristic (modal) rate. As I showed here, there is no consistent way to make this bidirectional. The Lorentz transformation is bidirectional. That is why the directional transformations are multiplied together in the derivation.
The Galilean transformation for one direction with spatial axis r and temporal axis t is:
Speed: r′ = r – vt and t′ = t,
Pace: t′ = t – ru and r′ = r.
A similar transformation for one direction with a characteristic rate c is:
Speed: r′ = r − tv = r (1 – v/c) and
t′ = t (1 – v/c) = t – (r/c) (v/c) = t – r (v/c²).
Pace: t′ = t – ru = t – r/v = t (1 – c/v) and
r′ = r (1 – u/b) = r (1 – c/v) = r − ct (c/v) = r − t (c²/v).
The Lorentz transformation (bidirectional with characteristic rate) is:
r′ = γ (r – tv) and t′ = γ (t – rv/c²) with γ = (1 – v²/c²)–1/2,
which applies only if |v| < |c|. The symmetric version is:
r′/c = γ (r/c – t (v/c)) = γt (1 – v/c), and
ct′ = γ (ct – r (v/c)) = γr (1 – v/c).
In the Lorentz transformation time is an independent variable; if space is independent:
r′ = γ (r − c2 t/v) and t′ = γ (t − r/v) with γ = (1 − c2/v2)–1/2,
which applies only if |v| > |c|. The symmetric version is:
r′/c = γ (r/c – t (c/v)) and ct′ = γ (ct – r (c/v)).