1) Independent variables are sometimes centered to reduce collinearity between higher order terms, as in the equation $$E(Y_i) = \beta_0 + \beta_1 X_{i} + \beta_2 X_{i}^2$$ with Y the explained variable
2) Independent variables can be normalized by dividing by their standard deviations. The interpretation of the coefficients is then "the expected change in Y when the predictor X is augmented by one standard deviation, all other variables kept equal". This standard deviation is known to the researcher, so he can recover the units. This approach renders quantitative comparison of coefficients of different variables possible, in terms of how they react to changes of the magnitude of their standard deviation.
3) Independent variables can be standardized by their range as you describe it. In this case the interpretation of the coefficients should be made in terms of "the expected change in Y when the predictor X is augmented by one observed range of this variable, all other variables kept equal". Again this range is known to the researcher, together with its units. At its minimum value this variable's contribution to the expected value will be 0, at its maximum equal to the coefficient.
