Surges are cool and dense ejections typically observed in chromospheric lines and closely related to other solar phenomena like UV bursts or coronal jets. Our aim is to address the current lack of inverted models and diagnostics of surges as well as characterizing their plasma properties. We have analyzed different surges related to UV bursts observed with the Interface Region Imaging Spectrograph (IRIS) on 2016 April. The mid- and low-chromosphere of the surges are examined by getting their representative Mg II h&k line profiles through the k-means algorithm and performing inversions on them using the STiC code. We have studied the far-UV spectra focusing on the O IV 1399.8 and 1401.2 Å lines, carrying out density diagnostics to determine the transition region properties of these ejections. We have also used experiments performed with the Bifrost code for comparisons. Thanks to k-means, we reduce the number of Mg II h&k profiles to invert by a factor 43.2. The inversions of the representative profiles show that the mid- and low-chromosphere of the surges have temperatures mainly around T=6 kK at −6.0≤log10(τ)≤−3.2. For the electronic number density, ne, and line-of-sight velocity, VLOS, the most reliable results from the inversions are within −6.0≤log10(τ)≤−4.8, with ne ranging from ∼1.6×10^-11 up to 10^12 cm^−3 , and VLOS of a few km s^−1. We find, for the first time, observational evidence of enhanced O IV emission within the surges, indicating that these phenomena have a considerable impact on the transition region even in the weakest far-UV lines. The O IV emitting layers show electron number densities from 2.5×10^10 to 10^12 cm^−3. The simulations provide theoretical support in terms of the topology and of the location of the O IV emission of the surges.