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emirp  and  nialpdrome

First numbers that belong to both sets:

31, 71, 73, 97, 311, 733, 743, 751, 761, 941, 953, 971, 983, 991, 3221, 7321, 7433, 7643, 9221, 9421, 9521, 9533, 9551, 9643, 9661, 9721, 9833, 9871, 9883, 9931, 9941, 72221, 73331, 74411, 74441, 75211, 75431, 75541, 76651, 77431, 77521, 77611, 77711, 94111, 94441, 95111, 96221, 96431, 96443, 96553, 97441, 97511, 97651, 97711, 98221, 98411, 98533, 98543, 98621, 98711, 98731, 98873, 98887, 99431, 99611, 99661, 99721, 99877, 99881, 721111, 732211, 733111, 733331, 733333, 742211, 743111, 743221, 754333, 755311, 755551, 764111, 765533, 766111, 766321, 766333, 775441, 776651, 777221, 777421, 777431, 777433, 777541, 777551, 777641, 777731, 777761, 911111, 955211, 965551, 966431, 966521, 966661, 972221, 973321, 973331, 974411, 974443, 975551, 976411, 976553, 977761, 982211, 984421, 985531, 986411, 986533, 986543, 986633, 988711, 988763, 994321, 995443, 995531, 995551, 996431, 996551, 997111, 997541, 997553, 997651, 997741, 997751, 998311, 998633, 998651, 998743, 998831, 998861, 999331, 999431, 999631, 999653, 999773, 999853, 999931, 999983, 3333311, 7441111, 7443221, 7444441, 7533221, 7541111, 7544431, 7554311, 7555211, 7555421, 7632211, 7642111, 7652111, 7653211, 7654441, 7655443, 7655521, 7661111, 7663211, 7664311, 7665431, 7666543, 7733321, 7744441, 7751111, 7753321, 7754321, 7754443, 7763311, 7764221, 7765553, 7766221, 7766621, 7772221, 7773211, 7773331, 7775533, 7777421, 7777543, 9331111, 9332111, 9433111, 9442211, 9543311, 9551111, 9554333, 9554431, 9555521, 9632111, 9654331, 9654443, 9665441, 9666311, 9666551, 9733211, 9743311, 9765211, 9765433, 9775553, 9776441, 9776651, 9777331, 9777661, 9777773, 9832111, 9854431, 9863221, 9863311, 9865411, 9865511, 9872111, 9874441, 9875321, 9875443, 9877633, 9886211, 9887333, 9887441, 9887611, 9887641, 9888443, 9888521, 9888533, 9888661, 9888733, 9888761, 9955441, 9955541, 9965311, 9965533, 9966511, 9966533, 9973331, 9974431, 9974443, 9976541, 9982211, 9984431, 9985111, 9987511, 9987533, 9988211, 9988633, 9988721, 9994333, 9994441, 9995333, 9995411, 9996211, 9997621, 9997643, 9998321, 9998543, 9998741, 9998887, 9999221, 9999433, 9999931, 9999943, 9999971, 33322111, 71111111, 74422211, 75432221, 75522221, 75544111, 75544433, 75554411, 75555551, 76433221, 76553311, 76554311, 76554431, 76654433.

Some larger examples :