倒しま is from the Nelsons. Most of my 漢和s don't have さかxx as a reading for 倒, apart from 広辞苑 which has さかさま. As you have seen, 広辞苑 has "さか‐しま【逆しま・倒】".

koj, daij, gg5, etc. all have 逆しま. "倒しま" may be kept as [io] if there's real evidence to support it from somewhere?

i don't think 倒しま should really be [news1,nf21].  prog,nc don't have it and meikyo says it's old-fashioned.  google hits suggest it may be based on an incorrect parsing of 倒します[たおします]

@@ -5,1 +5,1 @@
-<keb>倒しま</keb>
+<keb>逆しま</keb>
@@ -8,0 +8,3 @@
+</k_ele>
+<k_ele>
+<keb>倒</keb>
@@ -16,0 +19,1 @@
+<pos>&adj-na;</pos>
@@ -20,0 +24,7 @@
+<sense>
+<pos>&n;</pos>
+<pos>&adj-na;</pos>
+<gloss>unreasonable</gloss>
+<gloss>absurd</gloss>
+<gloss>wrong</gloss>
+</sense>