We perform an extensive statistical investigation of how interplanetary fast forward shocks affect certain turbulence parameters, namely, the normalised cross-helicity, σ_c; residual energy, σ_r; and magnetic helicity, σ_m. A total of 371 shocks detected by Wind at 1 au and 7 shocks by Solar Orbiter at 0.3–0.5 au have been analysed. We explore how the aforementioned turbulence parameters and their variation across the shock depend on the shock characteristics parameterised in terms of the gas compression ratio, upstream plasma beta, velocity jump, and shock angle. In the shock vicinity, fluctuations tend on average to show anti-sunward imbalance (measured as positive σ_c when rectified to the Parker spiral direction), a dominance of magnetic energy (negative σ_r) and zero σ_m, all being typical properties of the solar wind. Anti-sunward imbalance and equipartition (σ_r∼0) in the upstream is increasingly prevalent with increasing shock velocity jump and decreasing upstream beta and shock angle. Shocks with large velocity jumps and gas compression ratios have considerably more balanced (σ_c∼0) and more magnetically dominated fluctuations downstream than upstream. From upstream to downstream, we also find that the occurrence of time periods fulfilling strict criteria for Alfvénic fluctuations (AFs) usually decreases, while that of those meeting the criteria for small-scale flux ropes (SFRs) increases. The occurrence of AF-like periods peaks for quasi-parallel shocks with large velocity jumps and small upstream beta values. The occurrence of SFRs increases with an increasing gas compression ratio and upstream beta. The shocks observed by Solar Orbiter below 0.5 au display similar distributions of turbulence parameters and upstream-to-downstream changes to those detected at 1 au. These results are relevant for understanding turbulence and charged-particle acceleration at collisionless shocks.