Control XAML:
<UserControl x:Class="DPTest.Control" Name="Root">
    <UserControl.DataContext>
        <local:ControlViewModel Test="{Binding MyDependencyProperty, ElementName=Root, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay}" />
    </UserControl.DataContext>
    <StackPanel>
        <TextBox Text="{Binding Test, UpdateSourceTrigger=PropertyChanged}" />
        <Button Content="Click me to change ViewModel.Test value to 1234" Command="{Binding ClickCommand}" />
    </StackPanel>
</UserControl>

Control code:
namespace DPTest
{
    /// <summary>
    /// Interaction logic for Control.xaml
    /// </summary>
    public partial class Control : UserControl
    {
        public Control()
        {
            InitializeComponent();
        }

        public static readonly DependencyProperty MyDependencyPropertyProperty = DependencyProperty.Register(
            "MyDependencyProperty",
            typeof(string),
            typeof(Control),
            new FrameworkPropertyMetadata(string.Empty, FrameworkPropertyMetadataOptions.BindsTwoWayByDefault, OnPropertyChanged)
        );

        public string MyDependencyProperty
        {
            get { return (string)GetValue(MyDependencyPropertyProperty); }
            set { SetValue(MyDependencyPropertyProperty, value); }
        }

        private static void OnPropertyChanged(DependencyObject sender, DependencyPropertyChangedEventArgs e)
        {
            // never fired
        }
    }
}

ControlViewModel:

    class ControlViewModel : DependencyObject
    {
        public static readonly DependencyProperty TestProperty = DependencyProperty.Register(
            "Test",
            typeof(string),
            typeof(ControlViewModel),
            new FrameworkPropertyMetadata("start value", FrameworkPropertyMetadataOptions.BindsTwoWayByDefault, OnPropertyChanged)
        );

        public string Test
        {
            get { return (string)GetValue(TestProperty); }
            set { SetValue(TestProperty, value); }
        }

        public ICommand ClickCommand { get; set; }

        public ControlViewModel()
        {
            ClickCommand = new DelegateCommand(OnClickCommand);
        }

        private void OnClickCommand()
        {
            Test = "1234";
        }

        private static void OnPropertyChanged(DependencyObject sender, DependencyPropertyChangedEventArgs e)
        {

        }
    }

Test Window:

    <StackPanel>
        <local:Control MyDependencyProperty="{Binding Path=Text, ElementName=CheckDP, UpdateSourceTrigger=PropertyChanged}" />
        <TextBox Name="CheckDP" />
    </StackPanel>