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plaindrome  and  zygodrome

First numbers that belong to both sets:

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3333, 3344, 3355, 3366, 3377, 3388, 3399, 4444, 4455, 4466, 4477, 4488, 4499, 5555, 5566, 5577, 5588, 5599, 6666, 6677, 6688, 6699, 7777, 7788, 7799, 8888, 8899, 9999, 11111, 11122, 11133, 11144, 11155, 11166, 11177, 11188, 11199, 11222, 11333, 11444, 11555, 11666, 11777, 11888, 11999, 22222, 22233, 22244, 22255, 22266, 22277, 22288, 22299, 22333, 22444, 22555, 22666, 22777, 22888, 22999, 33333, 33344, 33355, 33366, 33377, 33388, 33399, 33444, 33555, 33666, 33777, 33888, 33999, 44444, 44455, 44466, 44477, 44488, 44499, 44555, 44666, 44777, 44888, 44999, 55555, 55566, 55577, 55588, 55599, 55666, 55777, 55888, 55999, 66666, 66677, 66688, 66699, 66777, 66888, 66999, 77777, 77788, 77799, 77888, 77999, 88888, 88899, 88999, 99999, 111111, 111122, 111133, 111144, 111155, 111166, 111177, 111188, 111199, 111222, 111333, 111444, 111555, 111666, 111777, 111888, 111999, 112222, 112233, 112244, 112255, 112266, 112277, 112288, 112299, 113333, 113344, 113355, 113366, 113377, 113388, 113399, 114444, 114455, 114466, 114477, 114488, 114499, 115555, 115566, 115577, 115588, 115599, 116666, 116677, 116688, 116699, 117777, 117788, 117799, 118888, 118899, 119999, 222222, 222233, 222244, 222255, 222266, 222277, 222288, 222299, 222333, 222444, 222555, 222666, 222777, 222888, 222999, 223333, 223344, 223355, 223366, 223377, 223388, 223399, 224444, 224455, 224466, 224477, 224488, 224499, 225555, 225566, 225577, 225588, 225599, 226666, 226677, 226688, 226699, 227777, 227788, 227799, 228888, 228899, 229999, 333333, 333344, 333355, 333366, 333377, 333388, 333399, 333444, 333555, 333666, 333777, 333888, 333999, 334444, 334455, 334466, 334477, 334488, 334499, 335555, 335566, 335577, 335588, 335599, 336666, 336677, 336688, 336699, 337777, 337788, 337799, 338888, 338899, 339999, 444444, 444455, 444466, 444477, 444488, 444499.

Some larger examples :