WPF - Left align items in an itemspresenter?

Question

I've got this bit of code in one of my control templates for a SplitButton I'm working on:

<Popup x:Name="PART_Popup"
       AllowsTransparency="True"
       Grid.Row="1"
       IsOpen="{Binding Path=IsDropDownOpen, RelativeSource={RelativeSource TemplatedParent}, Mode=TwoWay}"
       Placement="Bottom"
       PopupAnimation="{DynamicResource {x:Static SystemParameters.ComboBoxPopupAnimationKey}}"
       StaysOpen="False"
       MinWidth="{Binding Path=ActualWidth, RelativeSource={RelativeSource FindAncestor, AncestorType={x:Type wpfcustomcontrols:SplitButton}}}">

    <mwt:SystemDropShadowChrome Color="Transparent">
        <Border x:Name="DropDownBorder"
                Background="{DynamicResource {x:Static SystemColors.WindowBrushKey}}"
                BorderBrush="{DynamicResource {x:Static SystemColors.WindowFrameBrushKey}}"
                BorderThickness="1">
            <ItemsPresenter KeyboardNavigation.DirectionalNavigation="Contained"
                            SnapsToDevicePixels="{TemplateBinding SnapsToDevicePixels}" />
        </Border>
    </mwt:SystemDropShadowChrome>
</Popup>

The problem is the items within the ItemsPresenter (which are MenuItems) keep centering their content. I'm trying to figure out how to left-align their content instead... I have even tried doing something like this for the menuitems themselves:

<MenuItem>
    <MenuItem.Header>
        <TextBlock HorizontalAlignment="Left"
                   Text="Item 1" />
    </MenuItem.Header>
</MenuItem>

But, still the items center their content. What am I missing?