diff --git a/src/HalfIntegers.jl b/src/HalfIntegers.jl
index c47a8f0..ce50b29 100644
--- a/src/HalfIntegers.jl
+++ b/src/HalfIntegers.jl
@@ -23,6 +23,7 @@ export
     # Functions
     half,
     ishalfinteger,
+    ishalfodd,
     onehalf,
     twice
 
@@ -561,6 +562,37 @@ ishalfinteger(::HalfIntegerOrInteger) = true
 ishalfinteger(::AbstractIrrational) = false
 ishalfinteger(::Missing) = missing
 
+"""
+    ishalfodd(x)
+
+Test whether `x` is numerically equal to some half-odd-integer.
+
+# Examples
+
+```jldoctest
+julia> ishalfodd(3.5)
+true
+
+julia> ishalfodd(2)
+false
+
+julia> ishalfodd(1//3)
+false
+```
+"""
+ishalfodd(x::Number) = _isodd(twice(x))
+ishalfodd(x::Rational) = denominator(x) == 2
+ishalfodd(::Integer) = false
+ishalfodd(::Missing) = missing
+
+if VERSION ≥ v"1.7"
+    _isodd(x) = isodd(x)
+else
+    _isodd(x::Real) = isodd(x)
+    _isodd(n::Number) = isreal(n) && _isodd(real(n))
+    _isodd(x::AbstractFloat) = isinteger(x) && abs(x) ≤ maxintfloat(x) && _isodd(Integer(x))
+end
+
 @static if VERSION ≥ v"1.7.0-DEV.263"
     Base.range_start_stop_length(start::T, stop::T, len::Integer) where T<:HalfInteger =
         Base._linspace(float(T), start, stop, len)
diff --git a/test/customtypes.jl b/test/customtypes.jl
index a2617df..27fa117 100644
--- a/test/customtypes.jl
+++ b/test/customtypes.jl
@@ -62,6 +62,7 @@ Base.convert(::Type{One}, x::Int) = x == 1 ? One() : error("can't convert $x to
     @testset "Properties" begin
         @test isfinite(MyHalfInt(2))
         @test ishalfinteger(MyHalfInt(3/2))
+        @test ishalfodd(MyHalfInt(3/2))
         @test isinteger(MyHalfInt(3))
         @test !isinteger(MyHalfInt(3/2))
         @test iszero(MyHalfInt(0))
diff --git a/test/runtests.jl b/test/runtests.jl
index 9138633..9280a98 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -374,6 +374,24 @@ end
         @test !ishalfinteger(im)
     end
 
+    @testset "ishalfodd" begin
+        for T in (halfinttypes..., halfuinttypes..., :BigHalfInt)
+            @eval @test !ishalfodd($T(3))
+            @eval @test ishalfodd($T(5/2))
+        end
+        @test !ishalfodd(5)
+        @test !ishalfodd(big(-5))
+        @test ishalfodd(5//2)
+        @test ishalfodd(-3//2)
+        @test !ishalfodd(4//3)
+        @test !ishalfodd(2.0)
+        @test ishalfodd(-7.5)
+        @test !ishalfodd(2.3)
+        @test !ishalfodd(π)
+        @test !ishalfodd(ℯ)
+        @test !ishalfodd(im)
+    end
+
     @testset "isinteger" begin
         for T in (halfinttypes..., :BigHalfInt)
             @eval @test isinteger($T(3))
@@ -1690,6 +1708,7 @@ end
     @test onehalf(missing) === onehalf(Missing) === missing
 
     @test ishalfinteger(missing) === missing
+    @test ishalfodd(missing) === missing
 end
 
 include("ranges.jl")