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pseudoperfect n.  and  zygodrome

First numbers that belong to both sets:

66, 88, 222, 444, 666, 888, 1100, 1122, 1144, 1188, 2200, 2244, 2288, 3300, 3344, 3366, 3388, 4400, 4422, 4488, 5500, 5544, 6600, 6666, 6688, 7700, 7722, 7744, 7788, 8800, 8844, 8888, 9900, 9944, 9966, 11000, 11100, 11144, 11166, 22000, 22200, 22266, 22288, 33000, 33222, 33300, 33366, 33444, 33666, 33888, 44000, 44400, 44466, 55000, 55500, 55566, 55888, 66000, 66222, 66444, 66600, 66666, 66888, 77000, 77700, 77744, 77766, 88000, 88800, 88844, 88866, 99000, 99222, 99444, 99666, 99888, 99900, 99944, 99966, 99988, 110000, 110022, 110088, 111000, 111100, 111144, 111188, 111222, 111444, 111666, 111888, 112200, 112266, 112288, 113300, 113322, 113344, 113388, 114400, 114444, 114488, 115500, 115544, 115566, 116600, 116622, 116655, 116688, 117700, 117744, 118800, 118866, 118888, 119900, 119922, 119944, 119966, 119988, 220000, 220044, 220088, 221100, 221144, 221166, 222000, 222200, 222222, 222288, 222444, 222666, 222888, 223300, 223344, 224400, 224466, 224488, 225500, 225522, 225544, 225588, 226600, 226644, 226688, 227700, 227744, 227766, 228800, 228822, 228844, 228888, 229900, 229944, 330000, 330044, 330066, 330088, 331100, 331122, 331144, 331188, 332200, 332244, 332288, 333000, 333222, 333300, 333344, 333366, 333444, 333666, 333888, 334400, 334422, 334488, 335500, 335544, 336600, 336644, 336666, 336688, 337700, 337722, 337744, 337788, 338800, 338844, 338888, 339900, 339944, 339966, 440000, 440022, 440055, 440088, 441100, 441144, 442200, 442244, 442266, 442288, 443300, 443322, 443344, 443388, 444000, 444222, 444400, 444444, 444488, 444666, 444888, 445500, 445544, 445566, 445588, 446600, 446622, 446688, 447700, 447744, 448800, 448866, 448888, 449900, 449922, 449944, 449988, 550000, 550044, 550088, 551100, 551144, 551166, 552200, 552222, 552244, 552288, 553300, 553344, 554400, 554466, 554488, 555000, 555222, 555444, 555500, 555522, 555544, 555555, 555588, 555666, 555888, 556600, 556644, 556688, 557700, 557744, 557766, 557788, 558800, 558822, 558844, 558888, 559900, 559944, 559988, 660000, 660044, 660066, 660088, 661100, 661122, 661144, 661188, 662200, 662244, 662288, 663300, 663344, 663366.

Some larger examples :