horizontalScroller

gistfile1.txt

<TabControl Classes="horizontalScroller" />

HorizontalScroller.xaml

  <Style Selector="ScrollViewer.horizontalScroller">
    <Setter Property="Template">
      <ControlTemplate>
        <DockPanel>
          <RepeatButton x:Name="ScrollLeftButton"
                        DockPanel.Dock="Left"
                        Foreground="{DynamicResource SystemControlForegroundBaseHighBrush}"
                        Command="{Binding PageLeft, RelativeSource={RelativeSource TemplatedParent}}">
            <RepeatButton.IsVisible>
              <MultiBinding Converter="{x:Static avaConv:MenuScrollingVisibilityConverter.Instance}"
                            ConverterParameter="0">
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="HorizontalScrollBarVisibility" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Offset.X" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Extent.Width" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Viewport.Width" />
              </MultiBinding>
            </RepeatButton.IsVisible>
            <Path VerticalAlignment="Center"
                  HorizontalAlignment="Center"
                  Fill="{Binding #ScrollLeftButton.Foreground}"
                  Data="M0.222099 4.6098L4.67223 0.159755C4.77515 0.0567475 4.91255 0 5.05905 0C5.20555 0 5.34295 0.0567475 5.44588 0.159755L5.7736 0.487394C5.98685 0.700889 5.98685 1.04788 5.7736 1.26105L2.03672 4.99793L5.77775 8.73895C5.88067 8.84196 5.9375 8.97928 5.9375 9.1257C5.9375 9.27228 5.88067 9.4096 5.77775 9.51269L5.45002 9.84025C5.34702 9.94325 5.2097 10 5.0632 10C4.9167 10 4.7793 9.94325 4.67637 9.84025L0.222099 5.38614C0.118929 5.2828 0.062263 5.14484 0.0625882 4.99817C0.062263 4.85094 0.118929 4.71305 0.222099 4.6098Z"
                  Width="6"
                  Height="10" />
          </RepeatButton>
          <RepeatButton x:Name="ScrollRightButton"
                        DockPanel.Dock="Right"
                        Foreground="{DynamicResource SystemControlForegroundBaseHighBrush}"
                        Command="{Binding PageRight, RelativeSource={RelativeSource TemplatedParent}}">
            <RepeatButton.IsVisible>
              <MultiBinding Converter="{x:Static avaConv:MenuScrollingVisibilityConverter.Instance}"
                            ConverterParameter="100">
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="HorizontalScrollBarVisibility" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Offset.X" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Extent.Width" />
                <Binding RelativeSource="{RelativeSource TemplatedParent}"
                         Path="Viewport.Width" />
              </MultiBinding>
            </RepeatButton.IsVisible>
            <Path VerticalAlignment="Center"
                  HorizontalAlignment="Center"
                  Fill="{Binding #ScrollRightButton.Foreground}"
                  Data="M5.7779 4.6098L1.32777 0.159755C1.22485 0.0567475 1.08745 0 0.940948 0C0.794445 0 0.657048 0.0567475 0.554122 0.159755L0.226401 0.487394C0.0131509 0.700889 0.0131509 1.04788 0.226401 1.26105L3.96328 4.99793L0.222255 8.73895C0.119329 8.84196 0.0625 8.97928 0.0625 9.1257C0.0625 9.27228 0.119329 9.4096 0.222255 9.51269L0.549976 9.84025C0.652983 9.94325 0.790299 10 0.936802 10C1.0833 10 1.2207 9.94325 1.32363 9.84025L5.7779 5.38614C5.88107 5.2828 5.93774 5.14484 5.93741 4.99817C5.93774 4.85094 5.88107 4.71305 5.7779 4.6098Z"
                  Width="6"
                  Height="10" />
          </RepeatButton>
          <ScrollContentPresenter Name="PART_ContentPresenter"
                                  CanHorizontallyScroll="{TemplateBinding CanHorizontallyScroll}"
                                  CanVerticallyScroll="{TemplateBinding CanVerticallyScroll}"
                                  Content="{TemplateBinding Content}"
                                  Extent="{TemplateBinding Extent, Mode=TwoWay}"
                                  Margin="{TemplateBinding Padding}"
                                  Offset="{TemplateBinding Offset, Mode=TwoWay}"
                                  Viewport="{TemplateBinding Viewport, Mode=TwoWay}" />
        </DockPanel>
      </ControlTemplate>
    </Setter>
  </Style>
  <Style Selector="ScrollViewer.horizontalScroller /template/ RepeatButton">
    <Setter Property="Width" Value="24" />
    <Setter Property="Height" Value="24" />
    <Setter Property="Padding" Value="0" />
    <Setter Property="BorderThickness" Value="0" />
    <Setter Property="HorizontalAlignment" Value="Stretch" />
    <Setter Property="HorizontalContentAlignment" Value="Center" />
    <Setter Property="RenderTransform" Value="{x:Null}" />
  </Style>
  
    <Style Selector="TabControl.horizontalScroller">
    <Setter Property="Template">
      <ControlTemplate>
        <Border Margin="{TemplateBinding Margin}"
                BorderBrush="{TemplateBinding BorderBrush}"
                BorderThickness="{TemplateBinding BorderThickness}"
                Background="{TemplateBinding Background}"
                HorizontalAlignment="{TemplateBinding HorizontalAlignment}"
                VerticalAlignment="{TemplateBinding VerticalAlignment}">
          <DockPanel>
            <ScrollViewer HorizontalScrollBarVisibility="Auto"
                          VerticalScrollBarVisibility="Disabled"
                          Padding="6,0"
                          Classes="horizontalScroller"
                          DockPanel.Dock="{TemplateBinding TabStripPlacement}">
              <ItemsPresenter Name="PART_ItemsPresenter"
                              VirtualizationMode="Simple"
                              Items="{TemplateBinding Items}"
                              ItemsPanel="{TemplateBinding ItemsPanel}"
                              ItemTemplate="{TemplateBinding ItemTemplate}" />
            </ScrollViewer>
            <ContentPresenter Name="PART_SelectedContentHost"
                              Margin="{TemplateBinding Padding}"
                              HorizontalContentAlignment="{TemplateBinding HorizontalContentAlignment}"
                              VerticalContentAlignment="{TemplateBinding VerticalContentAlignment}"
                              Content="{TemplateBinding SelectedContent}"
                              ContentTemplate="{TemplateBinding SelectedContentTemplate}" />
          </DockPanel>
        </Border>
      </ControlTemplate>
    </Setter>
  </Style>

Actually I found this answer that works and is pretty close to what I want: AvaloniaUI/Avalonia#11896 (reply in thread)