tRNAs are widely believed to segregate into two classes, I and II. Computational analysis of eukaryotic tRNA entries in Genomic tRNA Database, however, leads to new, albeit paradoxical, presence of more than a thousand class-I tRNAs with uncharacteristic long variable arms (V-arms), like in class-II. Out of 62,202 tRNAs from 69 eukaryotes, as many as 1431 class-I tRNAs have these novel extended V-arms, and we refer to them as paradoxical tRNAs (pxtRNAs). A great majority of these 1431 pxtRNA genes are located in intergenic regions, about 18% embedded in introns of genes or ESTs, and just one in 3'UTR. A check on the conservations of 2D and 3D base pairs for each position of these pxtRNAs reveals a few variations, but they seem to have almost all the known features (already known identity and conserved elements of tRNA). Analyses of the A-Box and B-Box of these pxtRNA genes in eukaryotes display salient deviations from the previously annotated conserved features of the standard promoters, whereas the transcription termination signals are just canonical and non-canonical runs of thymidine, similar to the ones in standard tRNA genes. There is just one such pxtRNA(ProAGG) gene in the entire human genome, and the availability of data allows epigenetic analysis of this human pxtRNA(ProAGG) in three different cell lines, H1 hESC, K562, and NHEK, to assess the level of its expression. Histone acetylation and methylation of this lone pxtRNA(ProAGG) gene in human differ from that of the nine standard human tRNA(ProAGG) genes. The V-arm nucleotide sequences and their secondary structures in pxtRNA differ from that of class-II tRNA. Considering these differences, hypotheses of alternative splicing, non-canonical intron and gene transfer are examined to partially improve the Cove scores of these pxtRNAs and to critically question their antecedence and novelty.