The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to
x = ∫ v dt = x0 + vt
So for two observers at constant velocity in relation to each other we have
x′ = x + vt
with their time coordinates unchanged: t′ = t if their origins coincide.
The dual Galilean transformation is based on the definition of lenticity: w = dt/dx, which for constant lenticity leads to
t = ∫ w dx = t0 + wx
So for two observers at constant lenticity in relation to each other we have
t′ = t + wx
with their length coordinates unchanged: x′ = x if their origins coincide.
These transformations reinforce the proposition that time is not necessarily the independent variable, and so is best understood as measured by a stopwatch rather than a clock.