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plaindrome  and  untouchable n.

First numbers that belong to both sets:

2, 5, 88, 124, 146, 188, 238, 246, 248, 268, 288, 336, 448, 556, 668, 1116, 1128, 1134, 1146, 1148, 1168, 1222, 1236, 1246, 1248, 1256, 1258, 1266, 1288, 1346, 1348, 1388, 1566, 1578, 1588, 1888, 2226, 2228, 2248, 2258, 2338, 2356, 2366, 2388, 2446, 2568, 2588, 2788, 3356, 3366, 3378, 3388, 3448, 3488, 3556, 3588, 3666, 3788, 3888, 4488, 4588, 4668, 4678, 4688, 5556, 5568, 5578, 5588, 5666, 5668, 5778, 5788, 6688, 6788, 7788, 11136, 11146, 11166, 11168, 11226, 11234, 11236, 11248, 11266, 11268, 11336, 11338, 11358, 11366, 11368, 11378, 11456, 11458, 11466, 11478, 11488, 11556, 11568, 11666, 11688, 12224, 12234, 12236, 12238, 12248, 12258, 12288, 12358, 12368, 12444, 12448, 12456, 12468, 12558, 12566, 12568, 12588, 12666, 12668, 12688, 12788, 12888, 13358, 13388, 13588, 13668, 13778, 13788, 14466, 14468, 14488, 14568, 14578, 14588, 14678, 14688, 15566, 15678, 15778, 16688, 16778, 22224, 22228, 22236, 22244, 22256, 22268, 22288, 22358, 22466, 22556, 22566, 22588, 22888, 23348, 23366, 23368, 23468, 23488, 23556, 23588, 23778, 24468, 24556, 24566, 24568, 24588, 24666, 24668, 25688, 27788, 28888, 33336, 33366, 33448, 33456, 33468, 33478, 33556, 33568, 33666, 33668, 33688, 33888, 34446, 34466, 34556, 34558, 34566, 34568, 34578, 34788, 34888, 35558, 35566, 35568, 35588, 35688, 35778, 36666, 37788, 37888, 38888, 44448, 44458, 44466, 44468, 44556, 44558, 44568, 44678, 44788, 45566, 45666, 45788, 46678, 46888, 47788, 47888, 48888, 55556, 55568, 55588, 55688, 55778, 55888, 56668, 56678, 58888, 66666, 66668, 66778, 66888, 67788, 67888, 77778, 77788, 111116, 111126, 111134, 111148, 111156, 111158, 111166, 111168, 111222, 111234, 111236, 111248, 111288, 111336, 111366, 111368, 111378, 111388, 111456, 111558, 111566, 111568, 111588, 111688, 111888, 112222, 112228, 112258, 112268, 112356, 112358, 112368, 112388, 112446, 112448, 112466, 112556, 112566, 112568, 113334, 113336, 113348, 113378.

Some larger examples :