Take that frown and turn it around
JavaScript (ES6), 108 107 bytes

s=&gt;(S="⊢▷⋆∘○≀∧⍡⍢⍣⍤⍥⍨⍩⊣◁⋆∘○≀∨")[S.search(s[1]||s)+7]+(s[1]?'':':')


Demo



let f =

s=&gt;(S="⊢▷⋆∘○≀∧⍡⍢⍣⍤⍥⍨⍩⊣◁⋆∘○≀∨")[S.search(s[1]||s)+7]+(s[1]?'':':')

;[
  ":⊢", ":▷", ":⋆", ":∘", ":○", ":≀", ":∧",
  "⍡", "⍢", "⍣", "⍤", "⍥", "⍨", "⍩"
]
.forEach(s =&gt; console.log(s + ' → ' + f(s)))