The formula for Richardson extrapolation with central differences works because the central difference formula is an even function in h. This means that the power series in h contains only even powers of h.
The justification of Romberg integration involved the Euler Maclaurin summation formula. When stating the formula we observed that also here -- just like with Richardson extrapolation with central differences -- only even powers of h appeared. Thus the composite trapezoidal rule is an even function in h which explains the similarity in the extrapolation formulas for Romberg integration. In the sketch of the proof we defined the Bernoulli polynomials and the corresponding Bernoulli numbers.
A note on the Euler-Maclaurin summation formula is available