A numerical and statistical study is performed to describe the positive and negative local subgrid energy fluxes in the one-dimensional random-force-driven Burgers turbulence (Burgulence). We use a subensemble method to decompose the field into shock wave and rarefaction wave subensembles by group velocity difference. We observe that the shock wave subensemble shows a strong intermittency which dominates the whole Burgulence field, while the rarefaction wave subensemble satisfies the Kolmogorov 1941 (K41) scaling law. We calculate the two subensemble probabilities and find that in the inertial range they maintain scale invariance, which is the important feature of turbulence self-similarity. We reveal that the interconversion of shock and rarefaction waves during the equation's evolution displays in accordance with a Markov process, which has a stationary transition probability matrix with the elements satisfying universal functions and, when the time interval is much greater than the corresponding characteristic value, exhibits the scale-invariant property.