How can you two-way bind a checkbox to an individual bit of a flags enumeration?

Check your DataObject which binds to the CheckBoxes contains Department property has an INotifyPropertyChnaged.PropertyChanged called on its Setter?

You could use a value converter. Here's a very specific implementation for the target Enum, but would not be hard to see how to make the converter more generic:

[Flags]
public enum Department
{
    None = 0,
    A = 1,
    B = 2,
    C = 4,
    D = 8
}

public partial class Window1 : Window
{
    public Window1()
    {
        InitializeComponent();

        this.DepartmentsPanel.DataContext = new DataObject
        {
            Department = Department.A | Department.C
        };
    }
}

public class DataObject
{
    public DataObject()
    {
    }

    public Department Department { get; set; }
}

public class DepartmentValueConverter : IValueConverter
{
    private Department target;

    public DepartmentValueConverter()
    {
    }

    public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
    {
        Department mask = (Department)parameter;
        this.target = (Department)value;
        return ((mask & this.target) != 0);
    }

    public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
    {
        this.target ^= (Department)parameter;
        return this.target;
    }
}

And then use the converter in the XAML:

<Window.Resources>
    <l:DepartmentValueConverter x:Key="DeptConverter" />
</Window.Resources>

 <StackPanel x:Name="DepartmentsPanel">
    <CheckBox Content="A"
              IsChecked="{Binding 
                            Path=Department,
                            Converter={StaticResource DeptConverter},
                            ConverterParameter={x:Static l:Department.A}}"/>
    <!-- more -->
 </StackPanel>

EDIT: I don't have enough "rep" (yet!) to comment below so I have to update my own post :(

In the last comment Steve Cadwallader says: "but when it comes to two-way binding the ConvertBack falls apart", well I've updated my sample code above to handle the ConvertBack scenario; I've also posted a sample working application here (edit: note that the sample code download also includes a generic version of the converter).

Personally I think this is a lot simpler, I hope this helps.

Here's something I came up with that leaves the View nice and clean (no static resources necessary, no new attached properties to fill out, no converters or converter parameters required in the binding), and leaves the ViewModel clean (no extra properties to bind to)

The View looks like this:

<CheckBox Content="A" IsChecked="{Binding Department[A]}"/>
<CheckBox Content="B" IsChecked="{Binding Department[B]}"/>
<CheckBox Content="C" IsChecked="{Binding Department[C]}"/>
<CheckBox Content="D" IsChecked="{Binding Department[D]}"/>

The ViewModel looks like this:

public class ViewModel : ViewModelBase
{
  private Department department;

  public ViewModel()
  {
    Department = new EnumFlags<Department>(department);
  }

  public Department Department { get; private set; }
}

If you're ever going to assign a new value to the Department property, don't. Leave Department alone. Write the new value to Department.Value instead.

This is where the magic happens (this generic class can be reused for any flag enum)

public class EnumFlags<T> : INotifyPropertyChanged where T : struct, IComparable, IFormattable, IConvertible
{
  private T value;

  public EnumFlags(T t)
  {
    if (!typeof(T).IsEnum) throw new ArgumentException($"{nameof(T)} must be an enum type"); // I really wish they would just let me add Enum to the generic type constraints
    value = t;
  }

  public T Value
  {
    get { return value; }
    set
    {
      if (this.value.Equals(value)) return;
      this.value = value;
      OnPropertyChanged("Item[]");
    }
  }

  [IndexerName("Item")]
  public bool this[T key]
  {
    get
    {
      // .net does not allow us to specify that T is an enum, so it thinks we can't cast T to int.
      // to get around this, cast it to object then cast that to int.
      return (((int)(object)value & (int)(object)key) == (int)(object)key);
    }
    set
    {
      if ((((int)(object)this.value & (int)(object)key) == (int)(object)key) == value) return;

      this.value = (T)(object)((int)(object)this.value ^ (int)(object)key);

      OnPropertyChanged("Item[]");
    }
  }

  #region INotifyPropertyChanged
  public event PropertyChangedEventHandler PropertyChanged;

  private void OnPropertyChanged([CallerMemberName] string memberName = "")
  {
    PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(memberName));
  }
  #endregion
}

Thanks for everyone's help, I finally figured it out.

I am binding to a strongly typed DataSet, so the enumerations are stored as type System.Byte and not System.Enum. I happened to notice a silent binding casting exception in my debug output window which pointed me to this difference. The solution is the same as above, but with the ValueProperty being of type Byte instead of Enum.

Here is the CheckBoxFlagsBehavior class repeated in its final revision. Thanks again to Ian Oakes for the original implementation!

public class CheckBoxFlagsBehaviour
{
    private static bool isValueChanging;

    public static Enum GetMask(DependencyObject obj)
    {
        return (Enum)obj.GetValue(MaskProperty);
    } // end GetMask

    public static void SetMask(DependencyObject obj, Enum value)
    {
        obj.SetValue(MaskProperty, value);
    } // end SetMask

    public static readonly DependencyProperty MaskProperty =
        DependencyProperty.RegisterAttached("Mask", typeof(Enum),
        typeof(CheckBoxFlagsBehaviour), new UIPropertyMetadata(null));

    public static byte GetValue(DependencyObject obj)
    {
        return (byte)obj.GetValue(ValueProperty);
    } // end GetValue

    public static void SetValue(DependencyObject obj, byte value)
    {
        obj.SetValue(ValueProperty, value);
    } // end SetValue

    public static readonly DependencyProperty ValueProperty =
        DependencyProperty.RegisterAttached("Value", typeof(byte),
        typeof(CheckBoxFlagsBehaviour), new UIPropertyMetadata(default(byte), ValueChanged));

    private static void ValueChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
    {
        isValueChanging = true;
        byte mask = Convert.ToByte(GetMask(d));
        byte value = Convert.ToByte(e.NewValue);

        BindingExpression exp = BindingOperations.GetBindingExpression(d, IsCheckedProperty);
        object dataItem = GetUnderlyingDataItem(exp.DataItem);
        PropertyInfo pi = dataItem.GetType().GetProperty(exp.ParentBinding.Path.Path);
        pi.SetValue(dataItem, (value & mask) != 0, null);

        ((CheckBox)d).IsChecked = (value & mask) != 0;
        isValueChanging = false;
    } // end ValueChanged

    public static bool? GetIsChecked(DependencyObject obj)
    {
        return (bool?)obj.GetValue(IsCheckedProperty);
    } // end GetIsChecked

    public static void SetIsChecked(DependencyObject obj, bool? value)
    {
        obj.SetValue(IsCheckedProperty, value);
    } // end SetIsChecked

    public static readonly DependencyProperty IsCheckedProperty =
        DependencyProperty.RegisterAttached("IsChecked", typeof(bool?),
        typeof(CheckBoxFlagsBehaviour), new UIPropertyMetadata(false, IsCheckedChanged));

    private static void IsCheckedChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
    {
        if (isValueChanging) return;

        bool? isChecked = (bool?)e.NewValue;
        if (isChecked != null)
        {
            BindingExpression exp = BindingOperations.GetBindingExpression(d, ValueProperty);
            object dataItem = GetUnderlyingDataItem(exp.DataItem);
            PropertyInfo pi = dataItem.GetType().GetProperty(exp.ParentBinding.Path.Path);

            byte mask = Convert.ToByte(GetMask(d));
            byte value = Convert.ToByte(pi.GetValue(dataItem, null));

            if (isChecked.Value)
            {
                if ((value & mask) == 0)
                {
                    value = (byte)(value + mask);
                }
            }
            else
            {
                if ((value & mask) != 0)
                {
                    value = (byte)(value - mask);
                }
            }

            pi.SetValue(dataItem, value, null);
        }
    } // end IsCheckedChanged

    private static object GetUnderlyingDataItem(object o)
    {
        return o is DataRowView ? ((DataRowView)o).Row : o;
    } // end GetUnderlyingDataItem
} // end class CheckBoxFlagsBehaviour