I believe that you can use a trigger in the style to achieve this. You can place your image left aligned and in the trigger change that to right.
Something similar to:
<Style.Triggers>
<Trigger Property="ItemsControl.AlternationIndex" Value="1">
<!-- Change image position here -->
</Trigger>
<Style.Triggers>
More information about how to use AlternationIndex here.
Edit - Working Sample
<Style TargetType="ListViewItem">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="{x:Type ListViewItem}">
<DockPanel>
<Image Source="/WpfApplication;component/Images/TestImage.jpg" DockPanel.Dock="Left" x:Name="rowImage"/>
<TextBlock Text="Testing..." Background="{TemplateBinding Background}"/>
</DockPanel>
<ControlTemplate.Triggers>
<Trigger Property="ItemsControl.AlternationIndex" Value="1">
<Setter Property="DockPanel.Dock" TargetName="rowImage" Value="Right" />
</Trigger>
</ControlTemplate.Triggers>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>