Controller:

public class SampleController {
    
    Set<String> picklistValues = new Set<String> {'New York','California','Texas','Los Angeles','Chicago','San Francisco', 'Washington'};
    Public List<String> leftSelected {get;set;}
    Public List<String> rightSelected {get;set;}
    Set<String> leftValues = new Set<String>();
    Set<String> rightValues = new Set<String>();
    
    public SampleController(){
        leftSelected = new List<String>();
        rightSelected = new List<String>();
        leftValues.addAll(picklistValues);
    }
    
    public PageReference getSelect(){
        rightSelected.clear();
        for(String s : leftSelected){
            leftValues.remove(s);
            rightValues.add(s);
        }
        return null;
    }
    
    public PageReference getDeselect(){    
        leftSelected.clear();
        for(String s : rightSelected){
            rightValues.remove(s);
            leftValues.add(s);
        }
        return null;
    }
    
    public List<SelectOption> getDeselectedValues(){
        List<SelectOption> options = new List<SelectOption>();
        List<String> objList = new List<String>();
        objList.addAll(leftValues);
        objList.sort();
        for(String s : objList){
            options.add(new SelectOption(s,s));
        }
        return options;
    }
    
    public List<SelectOption> getSelectedValues(){
        
        List<SelectOption> options = new List<SelectOption>();
        List<String> objList = new List<String>();
        objList.addAll(rightvalues);
        objList.sort();
        for(String s : objList){
            options.add(new SelectOption(s,s));
        }
        return options;
    }
}

Visualforce Page:

<apex:page controller="SampleController">
    <apex:form>
        <apex:panelGrid columns="3" id="abcd">
            <apex:selectList id="sel1" value="{!leftSelected}" multiselect="true" style="width:100px" size="5">
                <apex:selectOptions value="{!DeselectedValues}" />
            </apex:selectList>
            
            <apex:panelGroup>
                <br/>
                <apex:image styleClass="picklistArrowRight" value="/s.gif">
                    <apex:actionSupport event="onclick" action="{!getSelect}" reRender="abcd"/>
                </apex:image>
                <br/><br/>
                <apex:image styleClass="picklistArrowLeft" value="/s.gif">
                    <apex:actionSupport event="onclick" action="{!getDeselect}" reRender="abcd"/>
                </apex:image>
            </apex:panelGroup>
            <apex:selectList id="sel2" value="{!rightSelected}" multiselect="true" style="width:100px" size="5">
                <apex:selectOptions value="{!SelectedValues}" />
            </apex:selectList>
        </apex:panelGrid>
    </apex:form>
</apex:page>

Output: