The standard relation between the cosmological redshift and cosmic scale factor is arguably one of a few unverified theoretical inputs underlying the computation of cosmological observables and modern cosmological inference. However, the richness of cosmological observations allows to change its theoretical status and the relation can now be directly measured from observations (hereafter redshift remapping in order to distinguish it from its theoretical counterpart). In my talk I will present the first non-parametric observational reconstruction of redshift remapping and explain its effect on the estimation of cosmological parameters. I will demonstrate that redshift remapping can be consistently determined in a dark-matter-dominated cosmological model. The resulting cosmological model resembles closely the standard LCDM model and has a number of intriguing properties, which I will discuss in the second part of my talk. I will also show that redshift remapping can shed light onto the problem of two persisting anomalies of the standard LCDM model: the discrepancy between the local and CMB-based Hubble constant and anomalous values of the BAO signal from the Ly-alpha forest of high redshift quasars.