We present optical spectroscopic follow-up of a sample of distant red galaxies (DRGs) with K ^tot _s,Vega < 22.5, selected by (J – K)_Vega>2.3, in the Hubble Deep Field South (HDFS), the MS 1054-03 field, and the Chandra Deep Field South (CDFS). Spectroscopic redshifts were obtained for 15 DRGs. Only two out of 15 DRGs are located at z < 2, suggesting a high efficiency to select high-redshift sources. From other spectroscopic surveys in the CDFS targeting intermediate to high-redshift populations selected with different criteria, we find spectroscopic redshifts for a further 30 DRGs. We use the sample of spectroscopically confirmed DRGs to establish the high quality (scatter in Δz/(1 + z) of ~0.05) of their photometric redshifts in the considered deep fields, as derived with EAZY. Combining the spectroscopic and photometric redshifts, we find that 74% of DRGs with K ^tot _s,Vega < 22.5 lie at z>2. The combined spectroscopic and photometric sample is used to analyze the distinct intrinsic and observed properties of DRGs at z < 2 and z>2. In our photometric sample to K ^tot _s,Vega < 22.5, low-redshift DRGs are brighter in K_s than high-redshift DRGs by 0.7 mag, and more extincted by 1.2 mag in A_V . Our analysis shows that the DRG criterion selects galaxies with different properties at different redshifts. Such biases can be largely avoided by selecting galaxies based on their rest-frame properties, which requires very good multi-band photometry and high quality photometric redshifts.